## Abstract

In this research, we present an interferometric system to analyze transparent samples using interferograms generated by a phase-shifting radial shear grating interferometer for two cases: the first obtaining $n$ simultaneous phase-shifting interferograms using a coherent light source and the second one using sequential phase steps with a white light source. For the first case, the simultaneous interferograms are generated using two optical systems: the first one generates the polarized pattern while the second one consists of a ${4}f$ system creating replicas of the output interferograms. By using a 2D sinusoidal phase grating, we have the advantage of obtaining up to nine replicated interferograms, all of them with comparable intensities and having amplitudes modulated by the 2D sinusoidal phase grating diffraction orders as zero-order Bessel’s functions. To obtain the optical phase map, several phase shifts are generated by placing a polarizing filter covering each replicated interferogram. We highlight the advantage of using $n$ simultaneous interferograms by comparing resulting optical phases processed by a conventional four-step algorithm against those obtained by an implemented ${n}={N}+{1}$ method, reducing errors with noisy interferograms. Results for ${n}={7}$ and ${n}={9}$ cases are presented. In addition, we have tested the setup with white light interference techniques by employing the polarizer radial shearing interferometer; for this case, the optical phase is calculated with the four-step and the three-step algorithms. Results of testing the developed system to examine static and dynamic phase objects are also included.

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

## Corrections

30 March 2020: A correction was made to Eq. (2).

## 1. INTRODUCTION

The advantages of radial shear interferometry (RSI) have been reported in optical testing for aberration measurements, aspherical surface analysis [1–4], adaptive optics applications [5–7], wavefront sensing [8–12], and beam characterization [13,14], among others. RSI systems have been implemented mainly with classical components [1,2,15–18] but also with gratings [19], zone plates, and holography [20,21]. Many of these applications benefit from the accuracy of the methods of phase-shifting interferometry (PSI) [22]. In previous reports, several interferometric systems were implemented as double window interferometers coupled to a ${4}f$ imaging system [23,24]. Such methods, where interferograms are generated by superimposing diffraction orders of the replicas of the two beams coming out from the interferometer, suffer the disadvantage of common polarization errors on the beams of the double window, in addition to systematic errors when calculating the separation of the double window according to the frequency of the grating and the focal length of the lens of the ${4}f$ imaging system. The system proposed here has the advantage of generating a base interferogram with known polarization properties [25–29], and the ${4}f$ system works as an interference pattern replicator. Several optical systems have been developed to retrieve the optical phase data from a single capture through polarization using micropolarizing arrays [26,27], grating interferometers [28], and also by liquid-crystal spatial modulators [30]. These systems are most commonly used to recover the optical phase in order to obtain the essential characteristics of the incident wavefront or physical properties of transparent samples. The primary purpose of this work is to measure variations of phase maps of transparent objects using several simultaneous interference patterns in a single camera capture. The system configuration is based on a polarized radial shear cyclic path interferometer (PRSCPI) that generates a base pattern with known polarization properties [25–29] and a ${4}f$ system with a 2D sinusoidal phase grating for replication purposes. The case of ${n}={4}$ interferogram replicas is widely used preferably when high-contrast and low-noise interferograms are available. On the other hand, when dealing with noisy patterns, algorithms with ${n}={7}$ or 9 might be used instead [31]. However, by acquiring interferograms sequentially, the PSI application is limited to static phase distributions. In this communication, we propose a radial shear interferometer capable of generating $n$ interferograms with appropriate phase shifts captured by a single shot. To generate the independent phase shifts, we used a grid of $n$ polarizers in the arrangement for each case, and the grid was built by cutting a polarizing sheet. The cut of the polarizers was done with a laser cutting machine; for the placement of the polarizers at the appropriate angles, a known oriented calibrated polarizer is used as an analyzer, and the transmitted intensities are measured. We checked the circular polarizations in every pattern by verifying that the intensity did not vary with the rotation of the analyzer, guaranteeing that the correct phase shift will be obtained, and the intensity of each pattern will be equal.

Our purpose is the development of phase dynamic interferometric configurations capable of analyzing phase variation in time. Since we are not employing specially manufactured components such as a phase cam with a micropolarizing array, high-speed cameras, or diffractive elements to retrieve the necessary interferogram replicas to achieve the measurements, the system uses polarization as a phase-shifting technique on each of the replicas by controlling a standard polarizer [32–34]. Some authors have used them for quantitative phase-imaging-based digital holographic microscopy, particle field holography, micro-metrology, among others [35,36]; these techniques have proven to be highly valuable in microscopic analysis. Such studies will be considered for future implementations of our approach; however, the quality of the results decreases due to the sample size and the diffraction effects due to the gratings [24,31]. Additionally, the radial shear systems are practical for use with white light because the difference between the two optical paths traversed by the radially sheared beams must be within one wavelength of the light [2,3,37]. The experimental results of a spherical wavefront for four, seven, and nine simultaneous interferograms, the case of a sample of red blood cells (RBC), the temporal variations caused by a thin flame and water fluid as an example of dynamic events and related experimental results using white light are presented.

## 2. INTERFEROMETRIC SYSTEM DESIGN AND BASIC PRINCIPLE

The schematic of the polarized phase-shifting interferometer is presented in Fig. 1 showing the usage of a telescopic system in PRSCPI to obtain radial shear in collimated light. The telescopic system consists of two lenses (${\rm L}_1$ and ${\rm L}_2$) with different focal lengths, where the incident collimated wavefront is taking two opposite paths around the interferometer. Thus, after completing the cyclical path in opposite directions, the emerging wavefronts are expanded by a factor $ M $ and contracted by a factor s, respectively (the radial shear distance $ M $ is the ratio of the beam diameters or magnifications, $M={f_2}/{f_1}$), resulting in a radial shear, where $ M $ is the ratio of the focal lengths of the two lenses. The transparent sample is placed on one arm of the interferometer (see Fig. 1). The two beams have different magnifications, the interference occurs over the common region to both beams, and the interference patterns are between the whole wavefronts and a magnified portion of its own center, producing interference fringes appearing over the entire aperture. The transparent sample is placed on one arm of the interferometer (see Fig. 1).

#### A. Radial Shear Polarizing Phase-Shifting Interferometry with a Sinusoidal Phase Grating

In the PRSCPI, the light passing after the first polarizer (${P_0}$) is linearly polarized at 45° and divided into horizontal and vertical linear polarization states after being reflected/transmitted with the polarizing beam splitter (PBS), that is, the PBS transmits the horizontally polarized beam and reflects the vertically polarized beam. The two beams collinearly counterpropagate in the cyclic path of the interferometer, and the beams recombine at the output of the PRSCPI and pass through the quarter-wave plate (QWP) placed at 45 deg with respect to the axial axis (the QWP makes circular polarization states in opposite directions). The radially sheared wavefront’s complex amplitudes can be presented by $O(x,y) = {a_0}( {x/M,x/M} ) \cdot \exp \{ { {i\phi ( {x/M,y/M} )} \}} $ and $R(x,y) = {a_1}( {M \cdot x,M \cdot y} ) \cdot \exp \{ { {i\phi ( {M \cdot x,M \cdot y} )} \}} $ having opposite circular polarization. Because the beams are circularly polarized, the amplitudes are comparable. Due to the different sizes of the beams, the beam size is filtered again, the same as with their common area. The amplitude ${P_b}(x,y)$ emerged from the output of PRSCPI is given by

where ${J_L}$ and ${J_R}$ are the Jones vectors of circular polarization to left and right, respectively, defined byIn this stage, the system does not generate an interference pattern because it has orthogonal polarization states. This can be seen in Eq. (1), $I(x,y) = {| {{P_b}(x,y)} |^2} = {\rm constant}$, in order to observe an interferogram, it is necessary to place a linear polarizer. For visualization purposes, we placed an auxiliary linear polarizer, and rotating it at any angle $\psi$ [see Fig. 1(a)], an interference pattern can be observed, maintaining a constant amplitude modulation [Fig. 1(b)] [26–29]. When each field is observed through a linear polarizing filter whose transmission axis is at an angle $\psi$, the output intensity considering these new polarization states is:

In the image plane of the completed system, the amplitude $T(x,y)$ can be written as

with $ \otimes $ denoting convolution. In the image plane of the system, a series of replicated patterns of the ${P_b}({x},{y})$ can be observed; this shown in Fig. 1(c). The pattern irradiance results are proportional to the squared modulus of Eq. (8) in the general case. Under these conditions, by detecting the irradiance with a polarizer array [Fig. 1(d)] at an angle $\psi$ with the horizontal [26–29], only the contribution of an isolated term of order $qr$ can be considered, and its irradiance would be proportional to## 3. OPTICAL PHASE RECOVERY ALGORITHM

#### A. Four-Step Algorithm

In previous sections, we have shown that the 2D sinusoidal phase gratings generate replicas of the interference patterns whose intensities are modulated by the intensities of the diffraction orders of the Bessel functions. Due to their comparable intensities, four simultaneous interference patterns were used [24]. It is necessary to place on each interferogram a linear polarizer with angles ${{\psi _1} = 0^\circ }$, ${{\psi _2} = 45^\circ }$, ${{\psi _3} = 90^\circ }$, ${{\psi _4} = 135^\circ }$ to obtain the respective phase shifts ($\xi$) of 0°, 90°, 180°, and 270° [23,24]; this is shown in Figs. 2(c) and 2(d), respectively. Thus, we can generate four simultaneous interference patterns spatially separated in the same image given by

#### B. Symmetrical (N + 1) Algorithm

To establish the advantages of the developed interferometric system, we analyzed seven and nine interferograms using the symmetrical (${N}+{1}$) phase steps algorithm for processing the cases ${n}={6} + {1}$ and ${n}={8} + {1}$ with a constant phase shift as a function of $ N $, where the ${N}+{1}$ interferogram results from a shift of 360°. The recovery optical phase formula for $ N $ shifts is given by [38–40]

For the case of symmetrical seven, each polarizer angle is ${\psi _1} = 0^\circ $, ${\psi _2} = 30^\circ $, ${\psi _3} = 60^\circ $, ${\psi _4} = 90^\circ $, ${\psi _5} = 120^\circ $, ${\psi _6} = 150^\circ $, and ${\psi _7} = 180^\circ $, which represent phase shifts ($\xi$) of 0°, 60°, 120°,180°, 240°, 300°, and 360°; this is shown in Figs. 2(e) and 2(f), respectively. For the case of symmetrical nine, each polarizer angle is ${\psi _1} = 0^\circ $, ${\psi _2} = 22.5^\circ $, ${\psi _3} = 45^\circ $, ${\psi _4} = 67.5^\circ $, ${\psi _5} = 90^\circ $, ${\psi _6} = 112.5^\circ $, ${\psi _7} = 135^\circ $, ${\psi _8} = 157.5^\circ $, and ${\psi _9} = 180^\circ $; in this case, each step ($\xi$) is 0°, 45°, 90°, 135°, 180°, 225°, 270°, 315°, and 360°, and this is shown in Figs. 2(g) and 2(h), respectively. This method reduces errors in phase calculations when noisy interferograms are involved [12]. Through Eqs. (11) and (12), the wrapped phase is obtained and the quality-guided path-following method was employed for the unwrapping process [39,40]. The amplitude modulation terms remain constant in all interferograms due to a normalization process that was applied to the fringe patterns to avoid possible errors introduced by small variations of the amplitude and the modulation in all interferograms. To remove the background phase, the phase retrieval procedure should include a reference phase which acts as a baseline measurement [28]; in other words, the background phase is calculated without the sample.

## 4. EXPERIMENTAL RESULTS

The laser used is an electrically driven diode-pumped solid-state laser that emits a pure TEM00 beam with diffraction-limited performance and a M2 of 1.05 (532 nm, 30 mW). The CMOS color camera used has a resolution of 3 megapixels (${2048}\;{\rm pixels} \times {1536}\;{\rm pixels}$) and frame rate of 12 fps; the individual patterns have a spatial resolution of ${140}\;{\rm pixels} \times {1140}\;{\rm pixels}$ and are treated as separate images by the implemented algorithm. For this purpose, we capture nine 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 is rather small if noticeable; this is an advantage of the phase-shift technique. To demodulate the phase, the interferograms retrieved by the optical system have a relative phase shift generated simultaneously and distributed in a rectangular image. Such a camera capture presents the interferograms distributed in the same image. The interferogram registration process starts by placing an iris diaphragm at the object plane of the interferometer to localize a common point on each interference replica. After locating each reference centroid of each point, a geometric mask that could be a rectangle, ellipse, or polygon, among other shapes, is selected in our program. Through this procedure, our program can select each ${n}$-interference pattern location. Subsequently, each of the patterns is cropped to generate independent images. This detection allows us to determine the number of steps captured by the camera. Finally, with the cropped patterns, the corresponding phase demodulation algorithm is employed on the interferograms and obtains the wrapped phase data. One of the disadvantages of this method is the generation of the polarizing array (PA) since the placement of each polarizer at the appropriate angle is a critical procedure for the generation of the correct phase shifts. To overcome this inconvenience, the polarizers were cut with a laser cutting machine LX-6090, and for their placement, the intensities were measured with a calibrated photodetector to ensure that the polarizers were placed at the correct angles.

The accuracy obtained in the measurement is typical of phase-shifting techniques. 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 if noticeable. Certainly, spatial resolution is compromised, and this can be noted when we obtain high-frequency fringes. This part can be improved by changing this aperture for a spatially known pattern and employing image registration techniques. Under our working conditions, we did not require any numerical compensation, and this will be considered for future implementations. The beam size of the collimated incident wavefront is 2 cm. It has been possible to experimentally verify that proper alignment and placement of the gratings allow us to obtain nine interferograms with comparable intensities. However, if the transmission axes of the gratings are not orthogonal or there are small displacements of the gratings, some grating orders display intensity losses. Figure 3 displays the results obtained when testing a wavefront. Figure 3 presents the case of four interferograms with a relative phase shift of $\pi /{4}$. Figure 3(a) shows the four simultaneous interferograms, and in Fig. 3(b) it is shown the calculated optical path difference (OPD) without removing the carrier. In Figs. 3(c) and 3(d) are shown the calculated OPD, which was fitted with a linear phase factor in order to remove the carrier. The use of ${n}={N}+{1}$ interferograms presents two main advantages of the conventional four-step method: First, it reduces errors in the phase estimation process when the interferograms have low signal-to-noise ratio; this is thanks to the high number of interferograms used. Second, it presents robustness in detuning errors due to the repetition of one step, so a second-order band-pass filter is applied in the frequency of the step [38–40]. Figures 4 and 5 shows the results obtained when testing a wavefront using the symmetrical ${N}+{1}$ phase step algorithms; for the cases of symmetrical seven (${6} + {1}$) and symmetrical nine (${8} + {1}$), the corresponding results and calculated phases are shown in Fig. 4 and Fig. 5, respectively. In both cases, we present in Figs. 4(a) and 5(a) the simultaneous interferograms, in Figs. 4(b) and 5(b) the calculated OPD without removing the carrier is shown, and in Figs. 4(c), 4(d) and 5(c), 5(d) the calculated OPD that was fitted with a linear phase factor in order to remove the carrier is shown.

In Fig. 6, we present the profiles of the noise of the obtained phases through the algorithms: 4 steps, ${6} + {1}$ steps, and ${8} + {1}$ steps. Figure 6(a) shows a cross section of the OPD for the optical phase shown in Figs. 3–5. The symmetrical (${n}+{1}$) algorithms can be used to reduce errors even when having noisy patterns, and algorithms with ${n}={7}$ or 9 might be used instead; see Fig. 6(b). Such profiles were obtained by subtracting a smooth curve that represents the shape of the phase. For display purposes, we added a bias of 1 and ${-}{1}$ in order to visualize the profiles. The standard deviations of each profile are ${\rm sigma}={0.068}\;{\rm OPD}[\lambda ]$ for the 4 steps, ${\rm sigma}={0.062}\;{\rm OPD}[\lambda ]$ for the ${6} + {1}$ steps, and ${\rm sigma}={0.056}\;{\rm OPD}[\lambda ]$ for the ${8} + {1}$ steps. This demonstrates that the larger the number of steps, the lower the signal-to-noise ratio.

The results obtained with a sample of red blood cells (RBC), placed by smears on a coverslip, are shown in Fig. 7. To process these results, the phase generated by the coverslip and the incident wavefront phase were first measured, and both phases were subtracted from the phase with the sample [28]. Figure 7(a) shows the four interferograms captured in one shot of the camera, and Figs. 7(b) and 7(c) present the OPD for all RBC under study.

Figure 7(d) shows the OPD obtained for the RBC enclosed in the dotted circle shown in Fig. 7(c). We can calculate the mean thickness as ${\rm OPD}/\Delta {n}={2.9}\;\unicode{x00B5}{\rm m}$, where the $\Delta {n}=1.39$ is the mean value of the RBC refraction index.

#### A. Dynamic Phase Measurements

In order to show the capability of the optical system to process dynamics events, Fig. 8 presents representative frames of a dynamic distribution of changing phase profile of the surroundings of a flame, with the calculated OPD (for this case, we used the symmetrical nine algorithm). The flame was placed in one arm of the interferometer; several frames were taken following the dynamic variation of the phase. In Video 1, the dynamic OPD evolution of the airflow around the flame is shown. The figure showed that the changes in the refractive index generated in the air around the flame of a candle could be observed. The results are obtained using an optical table without pneumatic suspension thanks to the stability properties of the common path interferometer implemented [15–18]. These results show that dynamic phase objects can be analyzed with the proposed optical system; see Visualization 1.

The representative frames of another dynamic event are shown in Fig. 9. The temporal variation of the radial slope associated with the phase deformations, generated by a water flow placed on a microscope slide, moving by gravity, can be observed [23,31]. The fringes obtained are contours of ${r}[\partial {r}({x},{y})/\partial {x}]$, and in Visualization 2 one can observe the phase variations induced by the fluid.

#### B. White Light Interference Fringes

White light interferometry is a powerful tool to perform optical measurements [41–44]. The optical setup was modified according to Fig. 10. In this case we used a white light source (Series Q Mercury Lamp, 50 W) that provides continuous-wave radiation and interference fringes that can be easily obtained. One of the advantages of employing cyclic path interferometers is the possibility of working in a zero-optical-path difference condition, and in our implementation, we employed the rectangular cyclic path configuration due to the possibility of using polarization phase-shifting techniques [2,3,37,41–45]. When using birefringent plates, which do not perform exactly as quarter-wave plates for the wavelength employed, the polarization angles of the linear polarizing filters to obtain 90° phase shifts must change according to the theory developed in the references [31,41].

However, the results reported here correspond to the acquisition of polychromatic phase-step interferograms in four stages; in this case, it is not a single-shot acquisition, since the phase steps are not obtained simultaneously. The optical phase of the incident wavefront can be processed using the four-step algorithm. The optical phase was processed by obtaining four polychromatic patterns with relative phase steps of $\pi /{2}$ (Fig. 11). Subsequently they are separated in the red, blue, and green channels, and each of them is processed separately to obtain the optical phase as shown in Fig. 12. The separated four $\pi /{2}$-phase patterns by channel are shown in Fig. 12(a), and their corresponding optical phases are shown in Fig. 12(b). The black line in Fig. 13 represents the cross section used to represent the intensities shown in Fig. 12. In Fig. 13, the cross section of the recovered phases is displayed, where it can be seen that the optical phase of the blue channel presents errors because the low response of the sensor of the used camera partially attenuates that wavelength.

Because the main objective is to obtain the optical phase with a single camera capture, in the following results (see Fig. 14), the case for a single capture is presented. One polychromatic interferogram was preprocessed to calculate the optical phase. The optical phase estimation for the color interferograms was processed by separating the channels of the image. Since the intensity of the patterns in each channel is different, depending on the sensitivity of the sensor of the camera, each pattern was normalized so the background is eliminated and the resulting amplitude of the signal is equal to 1 by using a preprocessing filter such as the ones presented in Refs. [46–48]. Finally, the wrapped phase is calculated with the well-known three-step algorithm [49]. It is noticeable that the resulting phase presents some detuning due to the nonuniform phase step between the channels; nevertheless, this can be fixed by applying a phase step estimator between the interferograms such as [50].

The polychromatic pattern is shown in Fig. 14(a), the separated normalized patterns are displayed in Fig. 14(b), and the OPD is shown in Figs. 14(c) and 14(d). The sample used is the edge of a transparent slide.

## 5. FINAL REMARKS

In conclusion, an adaptation of a radial shear interferometer with a 2D-phase grating to simultaneously achieve several interferograms for phase measurements using polarizing phase-shifting techniques has been demonstrated. Tests with $ 2\pi /N $ phase shifts and four interferograms were presented, but other approaches using different phase shifts could be attained using linear polarizers with their transmission axes at the proper angle before detection. Because of this, the presented system is capable of obtaining sets of ${n}$ interferograms with one single capture of the camera. We have tested white light interference techniques by employing a modification of this system; at the moment, no single-shot measurements had been done, and these techniques present possibilities to be used by controlling the spectral response of the system. This configuration is mechanically stable, and another advantage is that this interferometric system allows the capture of time-evolving dynamic events to be used for phase extraction.

## Funding

National Council of Science and Technology (A1-S-20925); Fondo Sectorial de Investigación para la Educación, Consejo Nacional de Ciencia y Tecnología.

## Acknowledgment

The authors thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper.

## Disclosures

The authors declare no conflicts of interest.

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