## Abstract

Light propagating through a scattering medium generates a random field, which is also known as a speckle. The scattering process hinders the direct retrieval of the information encoded in the light based on the randomly fluctuating field. In this study, we propose and experimentally demonstrate a method for the imaging of polarimetric-phase objects hidden behind a scattering medium based on two-point intensity correlation and phase-shifting techniques. One advantage of proposed method is that it does not require mechanical rotation of polarization elements. The method exploits the relationship between the two-point intensity correlation of the spatially fluctuating random field in the observation plane and the structure of the polarized source in the scattering plane. The polarimetric phase of the source structure is determined by replacing the interference intensity in traditional phase shift formula with the Fourier transform of the cross-covariance of the intensity. The imaging of the polarimetric-phase object is demonstrated by comparing three different phase-shifting techniques. We also evaluated the performance of the proposed technique on an unstable platform as well as using dynamic diffusers, which is implemented by replacing the diffuser with a new one during each phase-shifting step. The results were compared with that obtained with a fixed diffuser on a vibration-isolation platform during the phase-shifting process. A good match is found among the three cases, thus confirming that the proposed intensity-correlation-based technique is a useful one and should be applicable with dynamic diffusers as well as in unstable environments.

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

## 1. Introduction

Coherence and polarization are characteristics of randomly fluctuating fields [1–6] and can be tailored by making light pass through a turbid medium. Coherence is the correlation between the fluctuations at two different points, while polarization is related to the correlation between the two orthogonal electric field components at the same point [1]. The Stokes parameters are employed widely to describe the polarization properties of random electromagnetic beams and take into account the vector properties of light [3,7,8]. They have been determined experimentally [9]. The correlation, which can be considered as the average of the fluctuating field, can be used to characterize the statistical properties of the light field [10–12]. The correlation properties of a random field and the holographic principle have been used together to image objects hidden behind a scattering medium [13,14]. In 2003, Emil Wolf proposed the unified coherence and polarization theory [5]. The coherence and polarization of an electromagnetic beam can be represented using the 2 × 2 coherence-polarization (CP) matrix [15–20]. Generally, the elements of the CP matrix are complex quantities. A suitable interferometer must be used for measuring them experimentally [15–20]. A field-based interferometer can be used to measure the elements of the CP matrix, and the results can then be used to obtain the polarimetric information of the object lying behind a scattering medium [19]. However, this technique is susceptible to external vibrations and disturbances [18,19]. The complex elements of the CP matrix can also be measured using the Fourier fringe analysis method, which is based on speckle holography and the intensity correlation [9,20,21]. The filtering of the individual CP matrix elements from the intensity correlation requires the use of a quarter-wave plate (QWP) and polarizer before the detection of speckle pattern [20,21]. Combinations of the intensity correlation at four different orientations of the QWP are used to recover the complex CP matrix elements and generalized Stokes parameters (GSPs). The recovered CP matrix elements or the GSPs can then be used to obtain the nonstochastic information of the object from the random light [21]. In this study, we proposed and developed a new technique for recovering the polarimetric phase from the speckle pattern. The underlying idea is to combine speckle holography with polarization-based phase shifting during the intensity correlation measurements. We exploited the relationship between the two-point intensity correlation and the sum of the squares of the four elements of the CP matrix for a random field that is both stationary and ergodic in space. This allows us to replace the ensemble average with the spatial average when evaluating the correlation. Moreover, this approach neither requires a polarizer nor the mechanical rotation of the QWP prior to the intensity measurements and hence is more stable.

Phase shifting introduces changes in the off-diagonal elements of the CP matrix and hence modulates the interference pattern of the two-point intensity correlation. Conventional phase shifting based on the interference of complex fields requires accurate knowledge of the magnitudes of the phase shifts as well as a highly stable environment during the experimental tests. A traditional phase-shifting approach is limited in dynamic measurements because they are based on stable interference. Warger [22] and Millerd [23] proposed methods based on polarization interference to address the problem of dynamic phase measurements. These require simultaneous measurement of the four intensities or implement under suppression of the background phase.

In the presence of a dynamic scattering medium, the field will fluctuate, and conventional phase-shifting approaches cannot be used. However, the relationship between intensity correlation and the structure of the polarized source in the scattering plane can be exploited to develop phase shifting based on the intensity correlation. This method is helpful for recovering the phase and polarimetric information during the phase-shifting process even in the case of a dynamic (new) random scattering medium. This is done by performing phase shifting in the speckle holography and the intensity correlation. Moreover, the fact that the stability of intensity interferometers is higher than that of field-based interferometers makes the proposed phase-shifting approach, along with two-point intensity correlation, suitable for use in unstable environments.

In this study, we reconstructed the polarimetric phases of objects lying behind a scattering medium through two-point intensity correlation measurements. To develop this technique, we considered the interference between two randomly fluctuating fields formed by two independent polarized sources. Further, the intensity correlation is evaluated from the speckle interference pattern. In order to exploit the advantages of phase shifting based on the intensity correlation, a phase shift was introduced in one of the orthogonal polarization components of the test object. This allowed us to generate the polarization fringes while ensuring that only the off-diagonal elements of the CP matrix and hence the cross-covariance of the intensity fluctuations are modulated. The use of polarization-based phase shifting with the cross-covariance allows the off-diagonal elements of the CP matrix to be recovered. The back propagation of the complex coherence function representing the correlation between the orthogonal polarization components yields the structure of the polarized source in the scattering plane. To compare the suitabilities of the different phase-shifting methods mentioned in this work, we employed them to determine the cross-covariance and recover the polarimetric phase of various test objects. Besides a static scattering imaging, we also examined its validity in the case of a vibrating platform as well as with different scattering diffusers during the phase-shifting process. In contrast to conventional phase-shifting approaches, the proposed approach could recover the phase in the case of random scattering even under unstable conditions.

## 2. Theoretical analysis

The vector of a polarized field $E({{{\boldsymbol r}_1}} )$ can be represented as ${\hat{e}_x}{E_x}({{{\boldsymbol r}_1}} )+ {\hat{e}_y}{E_y}({{{\boldsymbol r}_1}} )$. Here, ${{\boldsymbol r}_1}$ is the position vector, ${E_x}({{{\boldsymbol r}_1}} )$ and ${E_y}({{{\boldsymbol r}_1}} )$ are the two orthogonal electric components, respectively, and ${\hat{e}_x}$ and ${\hat{e}_y}$ are the horizontal and vertical polarization states, respectively, of the light. This field is in a form wherein the polarization and spatial modulation are entangled inseparably [24]. Assuming that this source structure, $E({{{\boldsymbol r}_1}} ),$ is located in the diffuser plane, the field immediately after the diffuser is given by

The propagation of the electric field components of the polarized light from the diffuser plane to the observation plane for distance *z* is shown in Fig. 1. The complex amplitude in the observation plane can be represented as

Let us assume that the stochastic field is composed of the object and reference sources. These two sources are assumed to be statistically independent. The complex field in the observation plane is given as

where superscripts*O*and

*R*denote the object and reference sources, respectively.

Then, the intensity in the observation plane can be obtained from

To describe the phenomena of polarization and coherence together, the CP matrix [17,25,26], which characterizes the second-order correlation properties of the stochastic field, is used. The elements of the 2 × 2 CP matrix $\left( {\begin{array}{cc} {{W_{xx}}}&{{W_{xy}}}\\ {{W_{yx}}}&{{W_{yy}}} \end{array}} \right)$ are given as ${W_{ij}}({\Delta {\boldsymbol r}} )= \left\langle {{E_i}({\boldsymbol r} )\cdot E_j^\ast ({{\boldsymbol r} + \Delta {\boldsymbol r}} )} \right\rangle $, $i,j = x,y$. Here, the asterisk (*) denotes the complex conjugate while the angle brackets (< >) denote the ensemble average. The ensemble average is replaced by the spatial average for the spatially fluctuating random field. The elements of the CP matrix can be related to the two-point intensity correlation, that is, the fourth-order correlation of the random field, through the following relationship [15,25]:

**.**Here, $W_{ij}^O({\Delta {\boldsymbol r}} )$ and $W_{ij}^R({\Delta {\boldsymbol r}} ) $ are the complex coherence functions for the object and reference, respectively. The object coherence function $W_{ij}^O({\Delta {\boldsymbol r}} )$ in Eq. (7) is given by [18]

The reference coherence function in Eq. (7) is used to provide a constant “reference” wave that covers the support of $W_{ij}^O({\Delta {\boldsymbol r}} )$ and is given by

*a*is the radius of the circular aperture, which is very small.

Here, a phase shift is introduced in one of the orthogonal electric components ${E_i}({{{\boldsymbol r}_1}} )$ using a polarization discrimination device such as spatial light modulator (SLM). The polarization phase shift in one of the orthogonal polarization components in the diffuser plane is included in the off-diagonal elements of the CP matrix, ${W_{xy}}({\Delta {\boldsymbol r}} )$ and ${W_{yx}}({\Delta {\boldsymbol r}} ),$ as given in Eq. (7). On the other hand, the two diagonal elements of the CP matrix, ${W_{xx}}({\Delta {\boldsymbol r}} )$ and ${W_{yy}}({\Delta {\boldsymbol r}} )$, remain unmodulated in Eq. (7). We use the polarization-based phase shift and the fact that only the last two terms of Eq. (7) are modulated to recover the polarimetric phase from the speckle pattern.

The square of the modulus of the off-diagonal term ($i \ne j) $ in Eq. (7) can be expressed as

## 3. Experimental

The experimental scheme for the phase imaging of objects from the laser speckle is shown in Fig. 2. A linearly horizontally polarized He-Ne laser (HNL210L, Thorlabs) with a wavelength of 632.8 nm is spatially filtered by a microscope objective and pinhole, which are composed of a spatial filter (SF). The laser is then collimated by lens L_{1} with a focal length of 150 mm. The collimated beam is oriented at 45° with respect to the horizontal direction using a half-wave plate (HWP) and then divided into two arms by beam splitter BS. The beam reflected from BS illuminates a phase-only SLM with a pixel pitch of 8 µm (PLUTO-VIS, HOLOEYE Photonics AG, reflective type). The SLM only allows for modulation in the horizontal direction in order to subject the beam from the test object to different phase shifts. The beam transmitted from BS, called the reference beam, is reflected by mirror M. Next, both the beam from the SLM and that from the mirror pass through BS again and subsequently through lens L_{3}. Then, they arrive at different positions of the ground glass (GG), as shown in the inset in Fig. 2. On one hand, the light from the SLM is imaged on the plane of the GG with the help of lenses L_{2} and L_{3}. The width of the computer-generated phase object on the SLM is 4 mm; the size of the incident light on the SLM needs to be larger than the object to ensure that the object is fully illuminated. Therefore, the SLM is placed slightly away from the focal plane (300 mm) of the lens in order to increase the size of the illumination beam. On the other hand, mirror M introduces a tilt in the incoming beam, and the tiled beam is focused at an off-axis position on the GG using lens L_{3}. The GG (DG05-220, Thorlabs, thickness of 2 mm) acts as a scattering medium and introduces an isotropic random phase. Next, the composed speckle generated by the beam passing through the GG is recorded by camera CCD (AVT PIKE F421B), which has a resolution of 2048 × 2048 pixels and a pixel pitch of 7.4 µm. The GG is static during the recording of the speckle. However, we also performed tests for dynamic diffusers, which is implemented by replacing the diffuser with a new one during each phase-shifting step. The CCD camera is placed at a distance, z, of 185 mm from the diffuser, and there is no imaging device between the diffuser and the CCD plane.

## 4. Experimental results

During the experiments, the structure of the object was encoded into only one of the polarization components (horizontal) of the light and the other orthogonal (vertical) component remained unchanged. The two orthogonally polarized components propagated coaxially.

We used five phase-type samples namely, a smiling face, a butterfly, the Chinese letter “guang,” the letter “A,” and the number “2” for testing. These had different levels of complexity and sizes. The resultant speckles or random intensities recorded by the CCD camera in the case of the smiling face for phase shifts of 0, $\pi /2$, $\pi $, and $3\pi /2$ are shown in Figs. 3(a)–3(d), respectively. Each speckle pattern was composed of two independent sources and exhibited a different statistical distribution, as indicated by the different sizes of the speckle grains. The cross-covariance corresponding to the speckle intensities in Figs. 3(a)–3(d) are shown in Figs. 3(e)–3(h), respectively. Spatial averaging was applied as replacement of the ensemble averaging for evaluation of the cross-covariance of the experimentally recorded intensity and this cross-covariance is given in Eq. (6). The fringes of the cross-covariance of the intensity correlation can be observed in the bright area at the center.

The Fourier transforms of the cross-covariances in Figs. 3(e)–3(h) along with the phase shifts yielded the structure of the polarized source in the diffuser plane and its conjugate images. Figures 4(a)–4(e) show the Fourier spectra of the cross-covariances for the smiling face, butterfly, Chinese letter “guang,” letter “A,” and number “2.” The Fourier transform of the cross-covariance generates three terms: the spectrum, its conjugate, and a dc term. Because of the relationship described in Eq. (8), the coherence function can be related to the intensity distribution or the structure of the polarized source in the scattering plane. The central and other unmodulated terms related to the diagonal elements of the CP matrix are suppressed during the implementation of the polarimetric phase shift using the proposed approach. The separation of two diagonal conjugate images in Figs. 4(a)–4(e) is determined by the off-axis position generated by the reference beam, that is, by the tilt introduced by mirror M.

The source structure can be determined from the Fourier transform of the estimated cross-covariance. Therefore, we can obtain the phase of the test objects projected onto the SLM in the scattering plane. Here we apply the proposed phase-shifting technique to the Fourier spectrum of the cross-covariance. The proposed method is a new approach for recovering the polarized phase without mechanically rotating the orientation of the QWP [20,21]. This allows for high stability. Different phase-shifting methods can be employed to reconstruct the phase of the source structure. In the study, we used the following three phase-shifting techniques for phase recovery.

#### 4.1 Two-step phase shifting

We used two intensities to recover the phases of the various test objects. The first intensity corresponded to a phase shift of 0 (i.e., no phase shift) while the second one corresponded to a phase shift of $\pi /2$. The cross-covariances corresponding to the two intensities were ${C_1}$ and ${C_2}$, respectively. Based on the assumption that the amplitudes of the object beam and reference beam are the same, by taking the Fourier transforms of ${C_1}$ and ${C_2}$, one can determine the phase of the test object, including the phase of the reference, using the following relationship:

where $tan(\; )$ is the tangent function and $C_n^{{\prime}}$ is the Fourier transform of cross-covariance ${C_n}$. Here, $\varphi ^{\prime}$ is the phase of the test object relative to the reference. Further, suffix*n*represents the number of phase shifts. While the algorithm is simple for two-step phase-shifting, the background is difficult to eliminate [27]. This results in poor reconstruction quality.

#### 4.2 Four-step phase shifting

Four-step phase-shifting is the technique used most widely for phase recovery. The four intensities with phase shifts of 0, $\pi /2$, $\pi ,$ and $3\pi /2$ are needed for this method. The phase can be obtained from

*n*= 1–4) correspond to the four intensities with phase shifts of 0, $\pi /2$, $\pi $, and $3\pi /2$, respectively. Equation (12) is similar to that for conventional phase shifting [28]. Given that the back propagation of the cross-covariance ($C_n^{{\prime}}$) yields the structure of the source, we can determine the phase from $C_n^{{\prime}}$ (

*n*= 1–4).

#### 4.3 Eight-step phase shifting

The phase of the sample, ${\varphi _O},$ can also be obtained using another phase-shifting technique:

The imaging results for the objects in Figs. 4(a)–4(c) in the case of eight-step phase shifting are shown in Figs. 5(a)–5(c), respectively. It should be noted that the dc term and conjugate images in $C_\textrm{n}^{{\prime}}$ are also involved in the calculations and every two $C_\textrm{n}^{{\prime}}$ are performed a subtraction operation in Eq. (13). Usually, if the conjugated spectra overlap with the dc term, it is difficult to reconstruct the object. However, the subtraction operation performed in the phase-shifting method resolves this problem. This is the advantage of the phase-shifting technique [14,28]. In addition, the size of the two images in Figs. 5(a)–5(c) depends on the size of the object projected onto the SLM. And we can adjust the distance between the SLM and lens L_{2} to match the object size. Therefore, the resolution of the recovered phase is influenced by this distance. If the object size is too small, diffraction will influence the reconstruction, resulting in poor resolution.

Using the letter “A” in Fig. 4(d) as an example, we compared the reconstruction performances of the three phase-shifting techniques. Here, the phase of the object consisting of the letter “A” is $\pi /2 $ while that of the background is 0. The phase of the object as reconstructed using the two-step, four-step, and eight-step phase-shifting methods is shown in Figs. 6(a)–6(c). In the case of two-step phase-shifting, the reconstruction quality of the part close to the central dc term is relatively poorer than that of the other regions. Further, while we could obtain the profile of the letter “A” by two-step phase shifting, both the retrieval background and the phase value of the object are not accurate. The use of different phase-shifting techniques may yield reconstructed phases with different levels of gray. The variations in the color from blue to red in Figs. 6(a)–6(c) correspond to variations in the phase from -${\pi }$ to ${\pi }$. It should be noted that the reconstructed phases shown in Figs. 6(a) and 6(b), which were calculated using Eq. (11) and Eq. (12), respectively, were the same and that both included the phase of the reference beam [29]. The value of the recovered phase of the test object relative to the background in Figs. 6(a) and 6(b) is ${\pi }/2$, which is consistent with the change in the original phase of the object relative to that of the background. Therefore, these two methods only yielded the relative phase and not the absolute phase. On the other hand, the background phase value for the eight-step phase-shifting method is almost 0 and the phase of the object is consistent with the initial phase value of ${\pi }/2$. Therefore, the reconstruction quality for the eight-step phase-shifting method is better than those for the four-step and two-step phase-shifting methods. Finally, for all three phase-shifting techniques, the presence of noise would hinder phase recovery.

In order to highlight the stability of the proposed intensity-correlation-based imaging approach and compare it with that of conventional phase-shifting methods, we performed several tests both under varying environments and diffusers. We used the number “2” to test the performance of the intensity-correlation-based imaging technique based on the eight-step phase-shifting method. The results are shown in Fig. 7.

Figure 7(a) shows the recovery of the polarimetric phase of the number “2.” In this case, the test was performed at a vibration isolation platform with a fixed diffuser. Figure 7(b) shows the results for the same object but in the presence of external vibrations. In this case, the test was performed after stopping the active vibration isolation function of the optical platform and placing a vibrating mobile phone in close vicinity of the experimental setup. The reconstruction result in Fig. 7(b) is comparable to that in Fig. 7(a) and highlights the significant immunity of the proposed method to external vibrations. Figure 7(c) shows the phases recovered using different diffusers. The diffuser was replaced by a new one in each step during the eight-step phase-shifting process. The eight speckles corresponding to the eight-step phase-shifting process are independent random patterns, thus suggesting that this technique can be used even with quasi dynamic random medium. However, the speed of replacement (or rotation) of the diffuser has to be smaller than the recording time of the CCD camera. We calculated the Pearson correlation coefficient [30] of the reconstructed results in Fig. 7 by comparing them to the initial phase imposed on the SLM. The correlation coefficients for the phases in Figs. 7(a)–7(c) were 0.91, 0.88, and 0.78, respectively, thus confirming that the reconstruction quality in the case of a dynamic random diffuser and a vibrating platform was comparable to that for a static diffuser on a platform with vibration isolation.

## 5. Conclusions

In this study, we developed a novel technique for determining the polarimetric phase from laser speckle. The underlying principle of the technique was discussed, and tests were performed to both validate the method and highlight its advantages. We reconstructed the phases of five test objects using the eight-step phase-shifting technique and compared the reconstruction quality for three different phase-shifting techniques for the same test object. The experimental results indicated that eight-step phase shifting yields better reconstruction results than four-step and two-step phase shifting. In addition, the performance of the proposed method in the case of dynamic diffusers, the reconstruction quality for variable/dynamic and static diffusers was compared. We also tested the proposed method on a vibrating platform by adding an external vibration source. The polarimetric phase could be recovered faithfully under both dynamic conditions and in the presence of external vibrations, thus highlighting the superiority of the proposed technique over conventional phase-shifting holography.

## Funding

National Natural Science Foundation of China (11674111, 61575070); Fujian Province Science Funds for Distinguished Young Scholar (2018J06017); Fundamental Research Funds for the Central Universities (ZQN-PY209); Science and Engineering Research Board (EMR/2015/001613); Subsidized Project for Postgraduates’ Innovative Fund in Scientific Research of Huaqiao University.

## Disclosures

The authors declare no conflicts of interest.

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