Stokes correlation to estimate topological charge from the speckle pattern

The orbital angular momentum (OAM) of the vortex or helical beam, connected with the topological charge (TC) of the beam, has been used as an alternative degree of freedom to encode and decode the information in the optical signals. Many detection schemes have been established to determine the TC configurations and decode the information at the receiver end of the encoded information from the transmitter end for the free space. However, when the helical beam is transmitted through the scattering medium, its wavefront is spatially scrambled without any direct information about the incident helical beam. Here, we propose and experimentally demonstrate a new approach to estimate the TC of the helical beam from the non-imaged randomly scattered pattern. The technique utilizes Stokes parameters (SPs) of the random field to evaluate higher-order SPs fluctuation correlation. A complete theoretical model of the technique is presented and verified with numerical simulation and experimental tests. A good agreement is found between the simulation and experimental results.


Introduction
A light beam with a spiral phase structure exp(im ) and orbital angular momentum (OAM) of mℏ(where ℏ is a plank's constant) is referred to as a helical beam [1][2][3][4]. Term φ represents the azimuthal coordinates and m denotes the TC of the beam [2,3]. A helical beam possesses a central dark core in the amplitude distribution and this arises due to a phase singularity in the helical wavefront [3,4]. The OAM of the helical beam is a crucial parameter and has shown many interesting applications in optical communication, particle trapping, image processing, and optical testing [4][5][6][7]. Remarkably, channel strength and spectral efficiency in optical communications are enhanced by OAM encoding and mode multiplexing methods, since OAM modes with various m are theoretically immeasurable and orthogonal to one another in Hilbert space [5]. Hence, to decode the data with various OAM states at the receiver end in OAM communication, a precise estimation of the TC connected with the helical beam is of great indispensable. Till now, a variety of approaches have been established to detect the TC of the helical beam based on interference and diffraction. In the interferometric approach, the interference fringes are recorded to monitor the TC values after a helical beam interferes with a reference beam, such as a plane wave, or spherical wave [8][9][10][11]. However, interferometric systems require complicated optical setups and are sensitive to external disturbances. To overcome this issue, a helical beam was diffracted through a specially designed geometric aperture including a triangular aperture [12,13], annular gratings [14], and a translated single slit [15]. In a recent development, a gradually changing-period spiral spoke grating (GCPSSG) was introduced to monitor the magnitude and sign of the TC of the helical beam. The magnitude of TC was evaluated by enumerating the petal in the diffraction pattern and the sign was distinguished from the rotation of the twisted fringes with respect to the grating [16]. These techniques were implemented in free space.
On the other hand, the existence of a scattering media in the propagation path distorts the wavefront of the incident beam and plays a detrimental role in practical applications [17]. Propagation of the incident coherent beam in the presence of an inhomogeneous media scrambles the wavefront and generates a granular pattern i.e. speckle [18]. The speckle patterns don't have any direct analogize with the beam but the beam information is hidden in this pattern. The information-bearing properties of the speckle patterns were explored to develop different 1 3 13 Page 2 of 7 detection schemes for the recovery of incident helical beams. Researchers have used coherent light to generate speckle patterns and utilized information-bearing properties of this pattern to develop detection techniques for the recovery and estimation of TC of the helical beam [19][20][21][22][23][24][25] and in the examination of polarization properties [26]. In a recent development, the Hanbury-Brown Twiss effect was combined with off-axis holography to estimate the TC of the helical beam. A distribution of cross-covariance of the random intensity pattern was utilized to estimate the TC by enumerating the number of petals [27]. In addition, some significant detection techniques have been developed in the last decade to detect the TC and spiral phase profile of the helical beam with partially coherent light [28][29][30]. Recently, incoherent light has also been explored to develop detection schemes for the recovery of TC and spiral phase profile of the helical beam propagating through rough scattering layers [31][32][33].
In this paper, we propose a new approach to estimate the TC of an incident helical beam from the random light using higher-order Stokes parameters (SPs) correlation.
Here we present a new theoretical basis to estimate the TC of the incident helical beam propagating through the scatterer by applying higher-order SPs correlation. To develop a new theoretical basis, SPs of the random light are used and higher-order correlations between SPs fluctuations are evaluated. This provides a 4 × 4 correlation matrix that contains 16 elements. Out of these 16 elements only one element is considered to build a theoretical basis and subsequently apply for the estimation of the TC of the incident helical beam. To validate the proposed approach, we design a highly stable experimental geometry in a coaxial propagation of two orthogonal polarization states of the light and use this configuration to estimate the TC of the incident helical beam from the random light. A detailed theoretical model, numerical simulation, and experimental results are presented below.

Methodology
Let us consider an orthogonally polarized monochromatic light source. The x-polarization state of the beam is loaded with a spiral wavefront and the y-polarization state is a plane wave. These two orthogonal polarization states coaxially travel along the z-direction. The complex amplitude at the z = 0 plane is expressed as and U y (r) = A 0 (r). ê x and ê y represent unit vectors along the x and y-polarization direction, respectively, and r is the spatial position vector at the source plane.A m (r) , A 0 (r) indicate the amplitude distributions of the helical beam and non-helical beam. The term exp [im (r)] denotes the azimuthal phase structure over the transverse plane with a phase singularity in the heart of a wavefront. Now the beam transmits through a random scatterer and is detected at any arbitrary distance z from the source as represented in Fig. 1. The complex field at the detector plane is represented as where U(r) indicates the random field at the detector plane. FR is Fresnel transform to accommodate the propagation effect from the scattering plane to the detector plane located at distance z. (r) indicates the stochastic phase introduced by the diffuser, r is the transverse spatial position vector at the detector plane. The random field at the detector plane, situated in the Fresnel regime, and given as For the free space propagation, the Fresnel Kernel is represented as where λ denotes the wavelength of the incident beam and k = 2 represents wave number.
The SPs of the instantaneous random field is expressed as [34,35] The 2 × 2 identity matrix 0 and the three Pauli spin matrices are expressed as The SPs fluctuations with respect to their average value are expressed as where S p (r) is the SP at a special spatial point and < S p (r) > indicates its average value. The statistical features of the random light are characterized by correlations between the SPs fluctuations. All presumable pairs of the two-point SPs correlation are described by a 4 × 4 matrix C pq (r 1 , r 2 ) with elements where parenthesis ⟨.⟩ indicates the ensemble average. The 4th-order correlation can be explained in terms of the 2ndorder correlation of the Gaussian random field. Therefore the correlation matrix C pq (r 1 , r 2 ) is represented as (a, b = x, y and p = 0, 1, 2, 3) where W ad (r 1 , r 2 ) =< U * a (r 1 )U d (r 2 ) > are the elements of the 2 × 2 coherence-polarization (CP) matrix which characterizes the vectorial stochastic field [35,36] Therefore, all pairs of SPs fluctuations correlation are represented as [22,34] The element C 23 r 1 , r of the matrix is evaluated using Eq. (10) where A = W xy (r 1 , r )W * yx (r 1 , r ), and B = W * xx (r 1 , r ) W yy (r 1 , r ).
Let us explore the use of Eq. (13) in the estimation of the TC from the speckle pattern.
Substituting Eq. (3) into the SPs correlation results in the cancelation of the phase term outside the Fresnel propagation kernel as shown in Eq. (3) and hence this phase term is ignored in Eq. (15) [37]. The spatial stationarity is achieved at an arbitrary z and this allows to replace the ensemble averaging by spatial averaging [36,37]. Considering r 1 = r + ∆r and r 2 = r, the cross-covariance of the SPs at the detector C pq r 1 , r = ⎛ ⎜ ⎜ ⎜ ⎝ C 00 r 1 , r C 01 r 1 , r C 10 r 1 , r C 11 r 1 , r C 02 r 1 , r C 03 r 1 , r C 12 r 1 , r C 13 r 1 , r C 20 r 1 , r C 21 r 1 , r C 30 r 1 , r C 31 r 1 , r C 22 r 1 , r C 23 r 1 , r C 32 r 1 , r C 33 r 1 , r plane is expressed in terms of a two-point correlation function as explained in Eq. (13) and the two-point correlation function is expressed [22,37] as Using the relation ∫ exp − ik z (r −r 1 ) ⋅ r dr = (r −r 1 ). Therefore Eq. 2 aΔr∕ z , and W yy (Δr) ∝ 1.
Substituting the value of W xy (Δr), W * yx (Δr), W * xx (Δr), and W yy (Δr), into Eq. (13) Therefore Eq. (13) modifies as Equation (18) connects the source and Stokes correlation function at the detector plane and the distribution of crosscovariance helps to make a relation between topological charge and petals to estimate TC. The SPs correlations are now represented by the function of the difference between the spatial coordinates i.e. ∆r. The distribution of the coherence function forms petals due to the imaginary part Sin2(Δ ) of the cross-covariance function C 23 (Δr) . Equation (18) is capable to provide an estimation of the TC of an incident helical beam from the randomly scattered light.

Experiment
To validate the applicability of the proposed method, we design a non-interferometric experimental geometry to estimate the TC of the helical beam as shown in Fig. 2. A spatially filtered x-polarized He-Ne laser light with a wavelength of 632.8 nm is attenuated by a neutral density filter (NDF). The half-wave plate (HWP) at 22.5° is used to orient the input polarization of the incident beam into a diagonal polarization. A 50:50 beam splitter (BS) is inserted to divide the beam into two equal-intensity beams. A transmitted beam from the BS illuminates a phase-only reflective type spatial light modulator (SLM) with a pixel pitch of 8 μm and a resolution of 1920 × 1080 (Pluto from Holoeye). The SLM only allows to modulate the x-polarization state of the incident beam and the y-polarization state of the beam remains un-effected. The SLM is loaded with a computer-generated hologram to generate a helical beam in the x-polarization state. The y-polarization state contains a plane wave. These coaxially propagating beams propagate through the random diffuser (GG) and random scattering from the GG generates a polarization speckle, i.e. speckle with spatial polarization fluctuation [39]. The polarization speckle is analyzed through a quarter-wave plate (QWP) and a linear polarizer (LP). The combination of QWP and LP is used to measure the SPs of the scattered light. The intensity patterns I( q , p ) are captured by the charged coupled device CCD which is placed at 250 mm from the GG plane for four different combinations of QWP's fast and LP's pass axis orientation ( q , p ) with respect to the x-axis. The two SPs are measured from the captured intensity speckle patterns by considering the following equations [22]. The experimentally measured SPs are used to evaluate the correlation between SPs fluctuation using Eq. (8) and ensemble averaging is replaced by spatial averaging. This process is applied to evaluate cross-covariance C 23 (Δr) from the experimentally detected SPs as described in Eq. (18). Finally, the topological charge is estimated by counting petals as described in Eq. (19).

Results and discussions
The SPs are measured from the experimentally captured intensity speckle pattern for the helical beams with m = 1, 2, 3, and the results are represented in Fig. 3. Figure 3a Fig. 4a-c represent a distribution of polarization correlation function C 23 (Δr) for vortex with m = 1, 2, 3, respectively. Corresponding experimental results are represented in Fig. 4d-f. In Fig. 4a, d distribution of coherence function forms 4 petals for vortex with m = 1. The distribution of coherence function forms 8 and 12 petals for vortex with m = 2 and m = 3 are represented in Fig. 4b, e and c, f, , respectively. The complex coherence function of the experimentally detected SPs is examined by considering spatial averaging under the condition of spatial ergodicity at the observation plane. The benefits of spatial averaging had been applied currently in many correlation imaging methods with due importance in spatial statistical optics [40]. This spatial averaging is realized by taking a portion of the SP at the detector plane as a matrix S n (x, y) which denotes one realization of the SP of the random field. Here x and y are the pixel spatial coordinates and take values up to 300 × 300 pixels. The cross-covariance of the SPs for the different realization is indicated as Where, M represents the number of different realizations of the matrix S n (x, y) appearing due to pixel-by-pixel movement of the matrix S n (x, y) over the experimentally recorded SPs. We have used a random pattern of size 1000 × 1000 pixels and 2D scanning of S n (x, y) over the random pattern gives 700 × 700 different realizations for averaging. On the other hand, the average value of the Stokes ⟨ S p (r) ⟩ is obtained by taking the average over the spatial patterns M and represented as ∑ M

M
. A small deviation in the experimental results in comparison to the simulation is possibly owing to experimental constraints. Some weak residual modes for corresponding field distribution may appear due to the phase profile used to generate vortex using pixilated SLM [41]. On the other hand, limited size polarization optics in the experiment and background coherence W * xx (Δr), and W yy (Δr) in the Eq. (13) also affects the reconstruction quality.

Conclusions
We have proposed and experimentally demonstrated a new approach to estimating the TC of the helical beam propagating through the scattering medium. The SPs of the random light are utilized to evaluate the higher-order SPs fluctuation correlation and subsequently applied for the efficient estimation of the incident helical beam. The distribution of the cross-covariance renders the information of the TC of the helical beam. The feasibility of the proposed approach is evaluated by numerical simulation  and followed by the experimental demonstration for the estimation of TC of the helical beam. The proposed experimental configuration is robust and offers flexibility owing to non-interferometric. The applicability of the developed approach has been experimentally demonstrated to estimate helical beams with m = 1, 2, 3.