## Abstract

The recording and reconstruction of the Stokes parameter is of paramount importance for the description of the vectorial interference of light. Polarization holography provides a complete vectorial wavefront, however, direct recording and reconstruction of the hologram is not possible in a situation where the object is located behind the random scattering layer. The Stokes holography plays an important role in such situations and makes use of the Fourier transform relation between the Stokes parameters (SPs) at the scattering plane and the generalized Stokes parameters (GSPs) of the random field. In this paper, we propose and experimentally demonstrate the Stokes holography with the Hanbury Brown-Twiss (HBT) interferometer. We also propose and implement a lensless Fourier configuration for the Stokes holography. This permits us to reconstruct the wavefront from the GSPs at any arbitrary distance from the scattering plane. The application of the proposed technique is experimentally demonstrated for the 3D imaging of two different objects lying behind the random scattering medium. Depth information of the 3D objects is obtained by digitally propagating the generalized Stokes parameters to a different longitudinal distance. The quality of the reconstruction is assessed by measuring the overall visibility, efficiency, and PSNR of the reconstruction parameters.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Holography which records and reconstructs the wavefront of the light finds wide range of applications. Direct recording of the hologram by a detector and its numerical reconstruction, called as digital holography (DH), has brought the holography a big step forward [1]. Different geometries for the DH have been proposed in the past and significant among them are in-line, phase shifting, off-axis and the Fourier transform holography. It was demonstrated that a point-reference coherent source in the principal plane of the object generates a lensless Fourier-transform hologram which provides a high resolution in the wavefront-reconstruction imaging [2, 3]. This lensless Fourier transform configuration is useful at short, ultraviolet, x-ray wavelengths, imaging through diffusive medium and coherence waves [4–7]. A single Fourier transform operation is used to retrieve the information from the Fourier transform hologram and it plays an important role in real time applications. However, consideration of the complete wavefront, i.e. amplitude, phase and polarization, is important for full field description of the light field [8–13]. Extension of the DH to the vectorial domain is possible by recording holograms of the orthogonal components of the light [9]. In certain situations, however, interference effect does not manifest itself as intensity modulations. For instance, interference between an *x*-polarized object beam and a *y*-polarized reference beam generates only polarization modulation which can be highlighted by measuring the SPs. The SPs are important to characterize the light field and to describe the vectorial interference [11].

In many applications, it is required to image an object obscured by a scattering medium. When the object is hidden behind the random scattering media, it is difficult to apply the usual DH recording and reconstruction approach. Several techniques have been developed in order to image through scattering medium [11–20]. Random scattering scrambles the vectorial wavefront and makes it a spatially fluctuating polarized field. In this situation, correlation parameters such as coherence-polarization matrix and GSPs have been used to analyze the randomly polarized fields to recover the wavefronts [11-12, 21]. These parameters can be used to develop a novel imaging techniques by exploiting the statistical features of the random fields without resorting to any wavefront correction schemes. Recently, Stokes holography was developed to synthesize the GSPs structure in 3D space and the technique makes use of a Fourier transform relation between the Stokes fringes at the scattering plane and GSPs at the Fourier plane [11]. Polarized objects hidden behind the random scattering medium is encoded into the Stokes fringes and reconstructed as distribution of the GSPs in 3D. This is realized by applying the field based interferometer and evaluating the GPSs at the back focal plane of a Fourier transforming lens.

In this paper, we propose a new method to use intensity interferometer, i.e. HBT type for Stokes holography. The HBT approach permits to remove a Fourier transforming lens as required in the field based interferometer for the Stokes holography [11]. Making use of this feature, we design and develop a new lensless Fourier transform holography setup for the GSPs. This helps to achieve spatial stationarity of the random fields at an arbitrary distance *z* from the scattering plane and replaces the ensemble averaging by the space averaging of the random field. The GSPs also termed as Stokes vector wave follow the wave features in exact analogy to a coherent function of a scalar field [22]. We further make use of the lensless Fourier transform holograms of the GSPs to recover the desired GSPs for depth recovery of the objects encoded into the Stokes fringes. This is implemented by digital propagation of the GSPs (rather than mechanical scanning of the detector). To the best of our knowledge, this is first such attempt to exploit the interference of the GSPs to realize lensless Fourier transform hologram for the Stokes vector waves and apply the HBT type interferometer for 3D imaging of the polarized objects. The detailed theoretical basis and experimental results are discussed below.

## 2. Principle

Let us consider a polarized monochromatic light source at plane 1 as shown in Fig. 1. A coherent and polarized light passing through the scattering layer propagates down to the observation plane 2 located at a distance *z* from the source structure. The complex amplitude of the polarized field immediately after the scattering surface is represented as

_{$\left|{E}_{p}(\widehat{r})\right|$}and

_{${\varphi}_{p}(\widehat{r})$}are the amplitude and phase information of the polarized source at the scattering plane and

_{$\phi (\widehat{r})$}is the random phase introduced by the non-birefringent scattering medium. The transverse spatial coordinate at the scattering medium plane is represented by

_{$\widehat{r}$}and

_{$p=x,y$}represents two orthogonal polarization components.

The scattered field at a distance *z*, is given as

_{${k}_{z}(\widehat{r})=k\sqrt{1-{\left(\left|\widehat{r}\right|/f\right)}^{2}}$},

_{$k=\frac{2\pi}{\lambda}$}is wavenumber and

_{$\lambda $}is wavelength of the light and

*f*is the Fourier transforming length. The correlation of the randomly scattered field is evaluated under the assumption of stationarity and ergodicity in space and is given [11] by

_{${W}_{pp\text{'}}({r}_{1},{z}_{1};{r}_{1}+\Delta r,{z}_{2})$}represent two point correlation of the orthogonal polarization components

_{$p$},

_{$p\text{'}$},

_{$\Delta z={z}_{2}-{z}_{1}$},

_{$\Delta r={r}_{2}-{r}_{1}$}and

_{${\u3008\cdot \u3009}_{s}$}represents spatial averaging and the relation

_{$\int \mathrm{exp}\text{\hspace{0.17em}}\left[-i\frac{2\pi}{\lambda f}({\widehat{r}}_{2}-{\widehat{r}}_{1})\cdot {r}_{1}\right]\text{\hspace{0.17em}}d{r}_{1}}=\delta ({\widehat{r}}_{2}-{\widehat{r}}_{1})$}is used in Eq. (3). It is important to mention here that a Fourier transforming lens is used to achieve spatial stationarity of the random field to replace ensemble average by spatial averaging in [11, 12, 21]. The random field at the observation plane can be characterized by the GSPs, given [11, 14, 23–26] as

_{$\u3008\cdot \u3009$}is ensemble average which can be replaced by spatial averaging (rather than time) for the spatial ergodic field [21, 26]. The GSPs are transformed to the conventional SP for correlations at single point, i.e.

_{${r}_{1}={r}_{2}$}.

The GSPs can be expressed in terms of the SPs at the scattering plane using Eqs. (3) and (4) and given [11] as

_{${S}_{n}(\Delta r,\Delta z),(n=0,1,2,3)$}are 3D GSPs that show the reconstructed images from the random field while

_{${s}_{n}(\widehat{r})$}are the SPs of the source which may be in the form of polarization modulations, i.e. holographic or non-holographic The polarization modulations

_{${s}_{n}(\widehat{r})$}carry signature of the object located at any arbitrary distance from the scatterer. Equation (5) is a basic relation of the Stokes holography and states that complex valued object encoded into the Stokes fringes can be reconstructed as 3D spatial structure of the GSPs of the random field.

To experimentally test the Stokes holography, a polarization interferometer is designed and developed in [11]. This helps to simultaneously detect the orthogonal polarization components _{${E}_{p}(r,z)$}of the random fields and hence recover the GSPs. The field interferometer is successful in retrieving the complex field and evaluating the second order correlation but also sensitive to external disturbances. Here, we propose intensity correlation, i.e. HBT approach, for reconstruction of the GSPs in the Stokes holography. The HBT type interferometer is less susceptible to vibrations and evaluates the fourth order correlation of the random field. The fourth order correlation of the Gaussian random field is proportional to the modulus square of the second order field correlation [27–33]. The HBT approach for the Stokes holography permits us to make use of the interference of the GSPs from two independent sources and realize a lensless Fourier transform geometry for the Stokes waves. The new recording and reconstruction approach for the lensless Stokes holography is as follows.

Let us consider that a coherent polarized light field at the observation plane is coming from two independent sources located in the Fresnel domain in Fig. 1. The complex field at the plane 2 in the Fresnel region is given as

_{$G(r,\widehat{r})$}is a propagation kernel and given as

_{$\mathrm{exp}\text{\hspace{0.17em}}(ikz)/(i\lambda z)$}is ignored from further consideration as we are interested in observations at a fixed

*z*value. Using the above two relations along with the Fresnel kernel, the complex field is written as

_{$I(r)={\left|{E}_{x}(r)\right|}^{2}+{\left|{E}_{y}(r)\right|}^{2}$}. The intensity correlation can be explained in terms of the second order field correlation for the Gaussian random field. Under consideration of the intensity correlation, as in the coming section, we have ignored the common phase factor outside the integration in Eq. (8). Removal of this phase factor is significant in the intensity correlation based measurement for spatial stationarity [21]. Therefore, a GSP of the random field coming from two independent sources is represented as

_{${E}_{p}^{1}(\widehat{r})$}and

_{${E}_{p}^{2}(\widehat{r})$}, we are justified by taking the cross correlation of these two random fields

_{$\u3008{E}_{p}^{1*}(r){E}_{p}^{2}(r+\Delta r)\u3009\approx 0$}and therefore the GSP in RHS of Eq. (9) is represented as

_{$\int \mathrm{exp}\left(-\frac{ik}{z}({\widehat{r}}_{2}-{\widehat{r}}_{1}).r\right)}dr=\delta ({\widehat{r}}_{2}-{\widehat{r}}_{1})$}in Eq. (9). Similar relations can also be derived for the remaining GSPs by connecting GSPs with source SPs by a Fourier relation as

_{${S}_{n}({r}_{1},{r}_{2}),(n=0,1,2,3)$}are the GSPs of the random fields while

_{${s}_{n}^{1}(\widehat{r})$}and

_{${s}_{n}^{2}(\widehat{r})$}are the SPs (instantaneous-non averaged for the source) at the scattering plane. The GSPs at the observation plane can be represented as interference of two Stokes waves asEquation (12) can be used to record the lensless Fourier transform hologram of the Stokes waves

_{${\left|{S}_{n}({r}_{1},{r}_{2})\right|}^{2}={\left|{S}_{n}^{1}({r}_{1},{r}_{2})+{S}_{n}^{2}({r}_{1},{r}_{2})\right|}^{2}$}. To record this hologram, we follow Eq. (5) and generate a reference Stokes wave as follows

_{${S}_{n}^{2}({r}_{1},{r}_{2})={\displaystyle \int \delta (\widehat{r})\text{\hspace{0.17em}}\mathrm{exp}\text{\hspace{0.17em}}\left[-i\frac{2\pi}{\lambda z}\widehat{r}\cdot ({r}_{2}-{r}_{1})\right]\text{\hspace{0.17em}}d\widehat{r}}$}. Here, a point polarized source

_{$\delta (\widehat{r})$}is selected in order to generate a uniform reference Stokes wave covering the support of

_{${S}_{n}^{1}({r}_{1},{r}_{2})$}to record the lensless Stokes wave hologram. This helps to recover the GSPs of the object’s wave from a reference Stokes wave

_{${S}_{n}^{2}({r}_{1},{r}_{2})$}.

Let us turn to the generation of the Stokes waves hologram using the intensity measurements. Here, we introduce intensity correlation of the polarized speckle. The two point intensity correlation of the Gaussian random field is directly related to the second order field correlation and given [31, 33] as

_{$\Delta I(r)=I(r)-\u3008I(r)\u3009$}is an intensity fluctuation with respect to its mean value. Equation (13) emphasize that cross-covariance of the intensity is composed of contributions of modulus square of all four GSPs. Therefore, our objective is to retrieve the GSPs from the measured intensity correlation. We use Eq. (13) based on the HBT experiment to recover the GSPs from the intensity measurements.

In order to retrieve individual GSPs, we inserted an assembly of a quarter wave plate (QWP) with a linear polarizer (LP) before the observation plane 2. The QWP is oriented at angle _{$\theta $} with respect to the *x*-axis and a LP also oriented in the same direction as shown in Fig. 2. The complex field after the polarization elements is given as

*z*plane can be expressed in terms of the GSPs and given [34] as

_{$\theta $}), such that

_{${\theta}_{1}=0\xb0$},

_{${\theta}_{2}=22.5\xb0$},

_{${\theta}_{3}=45\xb0$}and

_{${\theta}_{4}=135\xb0$}four intensity correlation patterns are recorded. They are combinations of the GSPs as

_{${S}_{n}^{1}({r}_{1},{r}_{2})$}using the Fourier analysis technique [34]. Fourier transform of the intensity correlation hologram provides spectra, its conjugate and a dc term. The spectra is filtered and translated to the origin of the frequency coordinate. The inverse Fourier transform of the centrally shifted spectra and its appropriate combinations provide all the desired GSPs of the object field. The recovered GSPs at the Fourier plane can be digitally propagated to retrieve the Stokes fringes at the scattering plane using Eq. (5), and also reconstruct the complex valued polarized objects as spatial distributions of the GSPs in the 3D space.

## 3. Experiment

In order to demonstrate a lensless Stokes holography for the Stokes waves and realize the spatial averaging, we design an experimental setup for the proof of principle experiment as shown in Fig. 2. This consists of a system to encode a polarized object into the Stokes modulations _{${S}_{n}(\widehat{r})$} behind the scattering medium as explained in section 2, and experimental arrangements to implement a lensless Fourier transform Stokes holography and reconstruction with the HBT approach. The experimental scheme is designed to apply spatial averaging as a replacement of the ensemble averaging at a particular observation plane *z* and uses the recovered GSPs for digital propagation as described in the Eq. (5). This gives an advantage in the context of recovering GSPs spatial structures and hence reconstructing the 3D object structures encoded into the hologram without any mechanical *z* scanning of the detector. Detailed description of Fig. 2 is as follows.

A linearly polarized beam from a He-Ne laser of wavelength 633nm (Melles Griot 25-LHP-928-230) is oriented at 45° with respect to *x*-direction, spatially filtered with microscope objective O_{1} (20X, NA = 0. 40) and pinhole S (10μm) and subsequently collimated by lens L1 (f = 150mm). The collimated beam splits into two arms by a non-polarizing beam splitter BS1. The transmitted beam illuminates the spatial light modulator SLM (Holoeye LC-R 720, reflective type; pitch pixel = 20μm) which carries hologram of an object (H) and this hologram is projected at the scattering plane by a 4f imaging system with unit magnification. The digital hologram displayed on the SLM is a computer generated hologram (CGH) of off-axis objects and numerically recorded in the Fourier geometry as shown in Fig. 3. Figure 3(a) shows the formation of Fourier transform hologram encoding the object information and Fig. 3(b) shows composition of two objects placed at different longitudinal planes and their digitally generated FTH.

Light coming from the SLM have an arbitrary polarization and travels through the ground glass GG1 (DG20/120/MD, Thorlab, 3mm thickness) and gets randomly distorted and generates speckle pattern. The speckle pattern further propagates towards the charged coupled device (CCD) plane. The 45° linearly polarized light reflected from the BS1 and folded by mirror M1 is focused at the ground glass plate GG2 using MO2. The focal spot at GG2 is placed at an off-axis in order to generate a reference Stokes wave with uniform amplitude profile and a linear phase structure as described in Eq. (11). The speckle coming out of the GG2 propagates towards the CCD plane. Both the ground glasses are placed at distance *z* = 280mm from the CCD plane in order to make a lensless Fourier transform geometry for the Stokes waves as described in Eqs. (12) and (13). The random fields coming from two independent diffusers are combined by BS2 and coherently add to generate the resultant speckle for the orthogonal polarization components as described in Eq. (6). The light further passes through a QWP, which is rotated at an angle *θ* with *x* direction and filtered by a LP. The transmission axis of the LP is placed in the *x* direction and the resultant field is captured by a monochrome CCD camera. The camera has a 14-bit dynamic range with a resolution of 2750X2200 pixels and a pixel pitch of 4.54 micron (Prosilica GX2750). Both ground glasses are static during the recording of the intensity distribution of the resultant speckle field. The intensity is recorded for four different orientation angles of the QWP as described in the previous section and four intensity correlation holograms are digitally obtained from the experimentally recorded speckle patterns.

## 4. Results

We have applied this technique for different cases of objects and demonstrated 3D imaging as distribution of the spatial structure of the GSPs. In our first case, we consider a digital Fourier transform hologram (DFTH) of two longitudinally separated objects by a distance 50mm as shown in Fig. 3. And this DFTH is used as an object displayed at the SLM plane. Resultant speckle patterns for this case are shown in Fig. 4 for the four different orientations of the QWP.

The recorded intensity patterns are random which do not have any visual information. Spatially averaged two-point intensity correlation functions are digitally evaluated from the recorded speckle patterns that correspond to each of the QWP rotation, which are equivalent to _{$\Gamma (\Delta r,\theta )$}with _{$\Delta r={r}_{2}-{r}_{1}$}. Fourier analysis operation and appropriate combinations of processed spectra of the intensity correlation holograms provide GSPs.

To evaluate the quality of reconstruction, we introduce parameters like visibility, reconstruction efficiency and peak signal to noise ratio (PSNR). The visibility is defined as the extent to which the reconstruction is distinguishable from the background noise. It is given by the ratio of the average intensity level of object, to the background intensity level [18]

where,_{${V}_{obj}$}is the visibility of the object,

_{${I}_{{O}_{avg}}$}is the average intensity of the object and

_{${I}_{{B}_{avg}}$}is the average intensity of the background. Reconstruction efficiency 𝜂 is another parameter used to measure the quality of reconstruction [18]. It is given as the ratio of the object signal to the total power and calculated asPSNR is also one of the parameters to analyze the quality of reconstruction. PSNR is defined as the ratio of the maximum possible power of a signal to the power of corrupting noise that affects the quality [35]. PSNR in logarithmic decibel (dB) is calculated aswhere,

_{$MA{X}_{o}$}is the maximum possible pixel value of the object and

*MSE*is the cumulative squared error between the compressed and the original image.

Results of the GSPs at the plane *z* = 280mm are shown in Figs. 5(a)-5(d). These results can be used to digitally propagate the GSPs to different planes and results at plane *z* + ∆*z* = 330mm are shown in Figs. 5(e)–5(h). Phase structure of the reconstructed object _{${S}_{0}(\Delta r)$} is shown in Fig. 5(i). From the experimental results, it is clear that light field impinging the ground glass GG1 is predominantly linear. The calculated visibility (_{${V}_{obj}$}), reconstruction efficiency (𝜂) and the PSNR values for the images are shown in Fig. 5.

Reconstruction of another polarization hologram _{$s(\widehat{r})$} was carried out and results are shown in Figs. 6(a)-6(h). In this case, we consider an object ‘**2**’ encoded into the Fourier transforming hologram. Figures 6(a)-6(d) represent the corresponding GSPs of an object at *z* = 280mm plane. Whereas Figs. 6(e)-6(h) show the propagated GSPs at different depth *z* + ∆*z* = 310mm, calculated by using numerical beam propagation technique based on angular spectrum method. Figures 6(a)-6(d) show that the object displayed on the ground glass GG1 has a predominantly elliptical polarization. A phase structure of the reconstructed object _{${S}_{0}(\Delta r)$}is shown in Fig. 6(i) as distribution of phase of the GSPs. The calculated visibility (_{${V}_{obj}$}), reconstruction efficiency (𝜂) and the PSNR values for the images are shown in Fig. 6.

## 5. Conclusion

In this paper, we have experimentally demonstrated the imaging through scattering media by employing the Stokes holography with the Hanbury Brown-Twiss approach. The object information is encoded into the hologram and hidden behind the scattering media is successfully recovered from the polarization speckles by using a Lensless Fourier transform geometry for the Stokes waves and making use of the intensity correlation. This is possible by recovering the 3D-GSPs of the randomly polarized light fields. The demonstrated technique is well efficient in retrieving the information by reconstructing objects at their actual positions and provides a 3D complex field imaging facility. This technique can be used to retrieve anisotropic features of the objects lying behind the scattering media. Reconstruction results are quantitatively estimated by determining visibility, reconstruction efficiency and PSNR.

## Funding

Science and Engineering Research Board (SERB) (EMR/2015/001613); Indian Institute of Space Science and Technology.

## Acknowledgment

The authors acknowledge support from IIST.

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