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Abstract
Characterizing the orbital-angular-momentum (OAM) modes is critically important for OAM-encoded data transfer. However, traditional characterizing approaches rely on either complex and precise experimental configurations or complicated prior information processing. In these scenarios, the correlation features of OAM-dependent speckles from the scattering effect have received little attention. Here, we investigate the cross-correlation characteristics of the OAM speckles resulting from a scattering medium and propose an appealing alternative for spatial mode sorting and demultiplexing based on the OAM-dependent speckles. We demonstrate theoretically and experimentally that the cross-correlation operation between two different OAM-dependent speckles can uniformly derive an annulus pattern, whose dimension is determined by the absolute topological charge difference value between the two corresponding OAM modes. Based on this mechanism, the original coherent OAM modes can be easily sorted by implementing the cross-correlation operation between incoherently measured OAM-dependent speckles. To fully capitalize on the orthogonal feature of the OAM-dependent speckles, we also construct OAM mode demultiplexing experiments using a ground glass diffuser, where both 8-bit grayscale and 24-bit RGB OAM-encoded data demultiplexing are successfully demonstrated with superior error rates. Results show that the OAM-dependent speckles, previously treated as a nuisance for practical applications, can be surprisingly utilized as a competitive candidate for OAM mode sorting and demultiplexing, thus opening new directions in optical communication and information processing.
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1. INTRODUCTION
Optical vortex beams carrying orbital angular momentum (OAM) have been extensively studied in light–matter interaction based applications [1–6]. As a unique optical degree of freedom, the OAM of the vortex modes is not inherently constrained and thus can be used to increase the channel capacity in optical communication [7–11]. It is worth noting that, in these applications, the measurement and characterization of the complex optical field’s OAM spectrum are extremely important [12,13], especially in the demultiplexing process in OAM-based data communication systems. Typically, the phase-flattening technique has been recognized as a standard approach to measure and characterize the OAM spectrum, where the incident spatial modes to be evaluated are converted into the fundamental mode based on various phase-type components, such as a spiral phase plate, a fork hologram, or a $q$-plate [14]. Another kind of typical approach is based on the interferometry technique, where the OAM modes can be effectively separated using cascaded interferometers even at a single-photon level [15–19], or the OAM spectrum can be reconstructed through the inverse Fourier transform of the measured angular coherence function of the complex field using the interferometric method [20,21]. Besides, a spatial mapping technique has also been used to achieve the OAM spectrum. For instance, specific spatial modes can be mapped onto different regions by using a sophisticated hologram [22,23], or the azimuthal phase front beam can be converted into one with a transverse phase gradient by employing a Cartesian to log-polar transformation [24,25]. In addition, other approaches such as time mapping [26], intensity flattening [27], using a multiplane light conversion (MPLC) device [28,29], diffractive deep neural network (D2NN) [30,31], or employing rotational Doppler frequency shift [32] all aim to measure the OAM spectrum. However, the aforementioned techniques suffer from several weaknesses and disadvantages of different kinds, such as poor efficiency (e.g., the OAM mode is identified once each), requiring fine optical alignment (because of the incorporated interferometers), complex optical configurations (using numerous free-space components, especially for the phase-flattening technique and cascaded interferometers), and limited capability for a large number of spatial components.
On the other hand, the past few decades have witnessed the exciting progress of information extraction from speckle patterns that are achieved after light transmits through a scattering medium [33–36]. A scattering medium such as ground glass or multimode fiber has been employed as a dispersive element for high-resolution spectrometers [37], universal reconfigurable linear operators in optical computation [38], scattering lenses for both imaging through opacity [39,40] and light focusing [41], and multispectral imaging with a monochromatic camera [42]. In recent years, many efforts have been dedicated to investigating the diffused light behaviors of optical vortex beams. The random scattering effect has been exploited for spatial mode sorting assisted by the wavefront-shaping technique [43–46]. Besides, the transmission characteristics of Laguerre–Gaussian (LG) [47] and vector [48] vortex beams through turbid media are also investigated. However, in most circumstances, the random behaviors of the vortex beam transmitting through a scattering medium are too intricate to be manipulated and utilized. Commonly used techniques are based on the linear relationship between the incident vortex beam light and the diffused light field after transmitting through the scattering medium. Therefore, the priority is to elucidate the random behavior of the considered diffuser, which mainly falls into two typical approaches. One needs to measure the scattering transmission matrix (TM) in the complex field, which correlates the incident light with the diffused light [49,50], while the other one relies on the deep-learning-enabled method by ignoring any phase information and only analyzing the intensities of random speckle patterns [51–57]. However, both the TM calibration and the training process are prohibitively slow and computationally intensive, making them impractical for highly efficient applications.
Here, we focus on exploiting the characteristics of the incoherently measured random intensity patterns achieved after vortex beams transmit through a scattering medium. Theoretical analysis of the cross-correlation features of the OAM-dependent speckles is established based on the moment theorem for Gaussian statistics, for the first time to the best of our knowledge. In fact, the possibility of establishing the non-trivial mode-to-mode OAM correlations on the second order was described very recently [58,59], but the analysis has not been generalized to fourth-order field correlations so far. A cross-correlation annulus is surprisingly revealed, the radius of which is dependent on the topological charge difference between the involved two OAM modes. Based on the theory of OAM-dependent cross-correlation annulus, we conceive an appealing alternative to sorting out the spatial modes and verify its effectiveness by numerical calculations and experiments. Moreover, by leveraging the orthogonal nature of the OAM-dependent speckles, the incoherent random speckle basis is used to effectively measure the OAM spectrum of a multiplexed coherent light field. The experimentally achieved error rate for an 8-bit grayscale OAM-encoded data demultiplexing is larger than 38 dB for a ${100} \times {100}$ image with a single basis, while it is ${\sim}{33}\;{\rm dB}$ for each color channel of a 24-bit RGB OAM-encoded one. The proposed OAM-dependent speckles for spatial mode sorting and demultiplexing can present superior performance to traditional methods in terms of the cost, complexity, and time consumption in the measurement.
2. RESULTS
A. Theoretical Model of OAM-Dependent Speckle Correlation Based on the Moment Theorem for Gaussian Statistics
In typical scenarios of the OAM spectrum measurement, no appreciable wavefront distortion occurs before the vortex beam arrives at the receiver since the vortex beam carrying specific or multiplexed OAM values generally propagates in free space or a waveguide. Once the vortex beam transmits through a strong scattering medium, e.g., a piece of ground glass, the speckle pattern is generally considered a nuisance for effective information extraction. Contrary to general intuition that either deliberately prevents the scattering effect or fully calibrates the scattering TM, here, we propose to make use of the OAM-dependent speckles for effective OAM mode sorting and demultiplexing, as shown schematically in Fig. 1. The thin ground glass diffuser can impose random phase modulation to the incident vortex beam impinged onto it. Then, random speckle patterns can be obtained on the image plane, resulting from the interference between different light paths. Although the highly scrambled light beam is dramatically different from the incident light in terms of the intensity distribution, the effective encoded information is still preserved in the diffused light. This is attributed to the linear light propagation process in a complex scattering medium. The fundamental issue is how to distinguish the OAM component from the speckle patterns without calibrating the TM or applying the deep-learning-based training method.
Fig. 1. Schematic diagram of OAM-dependent speckles. Speckles are obtained when vortex beams carrying different topological charges transmit through a fixed ground glass diffuser.
Inspired by the works in speckle-correlated imaging [40,60–62] and optical speckle image velocimetry [63], we resort to analyzing the effective information concealed in the scrambled speckles by considering the cross-correlation feature of the OAM-dependent speckles. To elucidate this process fundamentally, the theoretical analysis of the cross correlation of the OAM-dependent speckles is conducted based on the moment theorem for Gaussian statistics. Here, to concentrate on the measurement of the OAM spectrum in the azimuthal direction and neglect its radial intensity variations for conventional vortex beams (e.g., LG modes) with an OAM-dependent radial structure, we aim to create a perfect vortex beam (PVB) based on a Gaussian apodization method as defined in the following model [64,65]:
where ${\rho _0}$ and $\Delta \rho$ represent the radius and width of the Gaussian-shaped annulus, respectively, and $\alpha$ denotes the topological charge. After passing through a diffuser (e.g., a ground glass plate), the two given vortex beams (i.e., $\alpha = l$, $m$) will be phase-modulated. The corresponding mutual coherence function (MCF) [66] right after the diffuser (i.e., the near-field regime) is given by expressionIn the far-field regime ($kz\; \to \;{\rm infinity}$, $k = {2}\pi /\lambda$, $k$ is the wavenumber, and $\lambda$ is the wavelength), in order to find the intensity cross correlation ${\langle}{I_l}({\vec r _1}^{\prime}){I_m}({\vec r _2}^{\prime}) \rangle $, the starting point is to acquire the MCF of the light field at two points (say, ${{ P}_1}$ and ${{P}_2}$) as given by expression [66]
Here ${\vec s _{1}^{\prime}}$ and ${\vec s _{2}^{\prime}}$ are unit vectors directed from the points ${{S}_1}$ and ${{S}_2}$ in the diffuser plane to the far-field points ${{P}_1}$ and ${{P}_2}$, respectively. The detailed notations can be found in Fig. S1 in Supplement 1 Note 1. With the assistance of expansion via the Bessel series, we can separate the integrals in the radial and angular variables yielding for the far-field same-point MCF in polar coordinates the expression
We can then deduce that $l - m = {n_1} + {n_2}$. Thus, the correlator exactly picks $l - m$ modes, assuring that only terms with $l - m$ can survive in the far field. This can be considered as the selection rule for the cross-correlation of the OAM-dependent speckles.
Fig. 3. Numerical simulations of the OAM-dependent speckles. (a1) Intensity and (a2) phase profiles of the PVBs, and (a3) corresponding OAM-dependent speckles for a fixed random phase modulation. (a4) Cross-correlation maps between the reference OAM-dependent speckle (${{I}_1}$) and the other ones (${{ I}_1}$, ${{I}_4}$, ${{I}_7}$, and ${{ I}_{10}}$). Xcorr, cross correlation. (b) Orthogonal relations between the OAM-dependent speckles corresponding to PVBs with topological charges from $\alpha = - {20}$ to $\alpha = {20}$ stepped by 1. (c) Calculated radius of the cross-correlation annulus as a function of topological charge when the reference OAM-dependent speckle corresponds to PVB with topological charge $\alpha = {1}$. TC, topological charge.
To solve the integral in the radial variable, we further simplify the ${I_n}(x)$ forms in Eq. (5) and introduce the confluent hypergeometric function (i.e., $M$-function) to deal with the integral of the Bessel function over the interval (0, $\infty$). The final expression of the far-field MCF can also be simplified by the connection between the $M$-function and the generalized Laguerre polynomials, as shown in Supplement 1 Note 1 in detail. Employing the deduced far-field MCF formula, the normalized intensity of the far-field MCF (i.e., ${| {{\Gamma _{\textit{lm}}^{(\infty)}}} |^2}$) is then calculated and provided in Fig. 2 (the required calculation parameters are also given in Supplement 1 Note 1). Here, the coordinate $\theta$ represents the angle that the line (from the point on the diffuser plane to the point on the far-field plane) makes with the positive $z$ axis. Four representative PVBs (e.g., $\alpha = {1}$, 4, 7, and 10) are considered. We make $m = {1}$ as the reference and $l = {1}$, 4, 7, and 10 for the cross-correlation calculation. It is shown that a cross-correlation annulus is revealed on the far-field plane and the radius is dependent on the value of $l - m$, which follows the selection rule predicted in Eqs. (6) and (7). Therefore, by transmitting the PVBs through a fixed random phase diffuser, the correlation is still concealed in the seemingly disturbed random speckles. We can simply figure out the individual topological charge of OAM modes based on the radius of the cross-correlation annulus obtained with the intensity cross-correlation operation. On finishing with the theory, we recall that during the derivation we set ${\rho _0} = {0}$ (for simplicity of calculations); i.e., the starting fields became the lowest-order Gaussian beams. Nevertheless, a ring-like cross-correlation intensity profile remains present, being a consequence of the correlation vortex after the diffuser plane, and not of the original PVB ring-like profile. It is also worth noting that the theoretical derivation process of the far-field MCF for the PVBs case can also apply to any other vortex beams passing through diffusers.
B. Numerical Simulations of the OAM-Dependent Speckles
The behaviors of the OAM-dependent speckles can also be analyzed numerically. The generation of the OAM-dependent speckles can be described by the light propagation model for the vortex beam transmitting through a scattering medium, as illustrated in Supplement 1 Note 2 [70]. The numerical codes of the PVB generation, random phase modulation, OAM-dependent speckle generation, and cross-correlation analysis are also provided in Supplement 1 Note 3. Numerical results of the OAM-dependent speckles corresponding to PVBs with topological charges ranging from ${-}{20}$ to 20 are calculated and given in Fig. 3. The intensity profiles of the PVBs with different topological charges maintain the same in terms of the radius and width of the annulus, as shown in Fig. 3(a1) for representative PVBs (e.g., $\alpha = {1}$, 4, 7, and 10). The period of the phase variation corresponds to the value of the topological charge, as shown in Fig. 3(a2). By imposing a fixed random phase modulation on the defined PVBs, scrambled speckle patterns (i.e., ${{ I}_\alpha}$) are then obtained on the image plane, as given in Fig. 3(a3), which leads to the direct loss of original annulus structural features of PVBs.
The Pearson correlation coefficients (the definition is given in Supplement 1 Note 3) among all OAM-dependent speckles are calculated in Fig. 3(b). The diagonal correlation feature confirms the orthogonality of the OAM-dependent speckles, which is inherited essentially from the orthogonality of original PVBs due to the linear transmission through the scattering medium [71]. On the other hand, the orthogonal OAM-dependent speckles are uncorrelated with each other and thus can be used as a set of discrete random bases. In this sense, we can simply assume that the OAM-dependent random bases are mapped from the original OAM ones by utilizing the ground glass diffuser.
The theoretical analysis confirms the OAM-dependent intensity correlation ring-like patterns in the far field of a diffuser, under the assumption of the ensemble averaging effect (typically obtained by diffuser’s rotation). Note that for a sufficiently strong diffuser, as is employed in this work, a single realization can be used instead of the ensemble. Then, it is of great necessity and significance to work out the actual alignment condition of the diffuser (e.g., a fixed position or a rapidly rotating one) to maintain the effective correlation transfer of the incident OAM modes. For a rapidly rotating diffuser, the long-time averaging effect on the speckles is involved. In contrast, a fixed diffuser alignment only allows for spatial averaging when taking the cross-correlation operation on the single-shot patterns. In Supplement 1 Note 4, we give detailed numerical calculation results for these two different diffuser alignments. The results show that, to retrieve effective cross-correlation information, both modes ($l$ and $m$) must go through the same diffuser’s realizations, whether for the single-shot or for the long-time ensemble averaging of intensity correlation. Therefore, the hidden correlation effect within the OAM-dependent speckles can be secured based on the same random phase realization for the OAM modes. Here, in this work, the intensity cross-correlation operation follows the relationship [61,63]
Figure 3(a4) displays the cross-correlation maps between the reference OAM-dependent speckle (${{I}_1}$) and the other ones (${{I}_1}$, ${{I}_4}$, ${{I}_7}$, and ${{I}_{10}}$). The result of cross correlation between the OAM-dependent speckle and itself, i.e., evolving into the autocorrelation operation such as $X{\rm corr}({{ I}_1},\;{{ I}_1})$, is a sharply peaked function as shown in Fig. 3(a4). However, for the OAM-dependent speckles corresponding to PVBs with different topological charges, a clear cross-correlation annulus appears on the cross-correlation maps. The radius of the cross-correlation annulus is determined by the absolute value of the topological charge difference, which is in line with the theoretical results. We also quantitatively calculate the mean value and the standard deviation of the radius of the cross-correlation annulus as a function of topological charge, where the reference OAM-dependent speckle corresponds to $\alpha = {1}$, as shown in Fig. 3(c). It is observed that the radius of the cross-correlation annulus almost increases monotonically with the topological charge difference. By setting a prior OAM mode with a given topological charge and measuring the corresponding OAM-dependent speckle pattern as a reference, it is theoretically possible to sort the other OAM modes just from the cross-correlation annulus between the OAM-dependent speckles.
Fig. 4. Experimental diagram of OAM-dependent speckles for spatial mode sorting and demultiplexing. DMD, digital micromirror device.
It is worth noting that in the theoretical model based on the moment theorem for Gaussian statistics, the cross-correlation annulus only appears in the far-field regime, not the near-field one (e.g., right after the diffuser). For a conventional optical system, the strict definition of the far-field condition generally requires meter-level propagation distance. However, it has also been demonstrated that the far-field condition can be dramatically shortened to several centimeters in a speckle correlography [72] since the speckle pattern has a small correlation radius. Here, we also numerically verified that the effective intensity cross-correlation can be realized for speckles monitored at the plane several centimeters away from the diffuser, as discussed in Supplement 1 Note 5. On the other hand, it is also known in Fourier optics that the far-field pattern can be obtained by taking the Fourier transform on the near-field one. To further verify the cross-correlation behavior of the OAM-dependent speckles, the Fourier transform is imposed on the near-field speckle field to numerically calculate the far-field speckle, as also discussed in Supplement 1 Note 5. The results show that the cross-correlation annulus is also revealed for the calculated far-field speckles and holds the same property for effective mode sorting. This validates the effectiveness of the proposed theoretical model and also indicates that the near-field speckle patterns recorded at a certain propagation distance are well qualified for the effective cross correlation. Therefore, for our numerical analysis and experimental verification, the image distance is also set to be several centimeters (e.g., 7 cm as shown in the numerical code in Supplement 1 Note 3). This feature can also alleviate the experimental complexity and enhance the system’s robustness.
C. Experimental Results of the OAM-Dependent Speckles
To experimentally analyze the characteristics of the OAM-dependent speckles, a super-pixel wavefront-shaping technique [73–75] is employed to generate the PVB and apply different values of OAM onto the beam as schematically shown in Fig. 4. Here, a homemade fiber laser (central wavelength of ${\sim}{1064}\;{\rm nm}$, 3 dB bandwidth of 0.35 nm) is used as the effective light source for the PVB generation. The output port of the fiber laser source is mounted on a two-dimensional translation stage and further collimated by a convex lens (focal length of 6.2 mm). The collimated light beam is then impinged onto the surface of a digital micromirror device (DMD, DLP650LNIR, Texas Instruments) via a highly reflective mirror. Binary patterns are generated by the super-pixel wavefront-shaping technique and then loaded onto the DMD. Combined with a ${4}f$ imaging system where an iris diaphragm is deployed on the Fourier plane to filter out the first-order diffraction light, the incident fundamental Gaussian mode is converted into the desired PVBs. The obtained PVBs are then incident onto a piece of ground glass (grit size of 120, Thorlabs), and the OAM-dependent speckles are relayed onto a charge coupled device (CCD, DCU224M, Thorlabs) camera.
The cross correlation of the OAM-dependent speckles is experimentally investigated, as shown in Fig. 5(a). It is worth noting that in the super-pixel wavefront shaping, the target intensity and phase are generated as that in the numerical simulation part in Supplement 1 Note 3. The radius $\rho_{0}$ and the width of the Gaussian-shaped annulus $\Delta \rho$ are 1.80 and 0.83 mm. Then the DMD binary hologram patterns are created through a pre-calculated lookup table. Here, the reference OAM-dependent speckle corresponds to PVB with a topological charge of $l = - {8}$, while the other ones correspond to $m = - {8}$, ${-}{5}$, ${-}{2}$, 2, 5, and 8, respectively. Apart from the sharp peak profile shown in the autocorrelation map of $m = - {8}$, a cross-correlation annulus can be discerned on the cross-correlation map, the radius of which depends on the topological charge difference. The normalized central cross-correlation curves corresponding to the cross-correlation maps (${Y}$ shift direction) in Fig. 5(a) as a function of $\theta$ are also provided in Fig. 5(b). The $\theta$ value is approximated by the ratio between the radial dimension on the cross-correlation map and the longitudinal propagation distance. The OAM-dependent annulus agrees well with both the theoretical analysis in Fig. 2 and the numerical analysis in Fig. 3(a4). The gradually pronounced background noise in the cross-correlation map [Fig. 5(a)] and the normalized central cross-correlation curves [Fig. 5(b)] can be attributed to the low-pass filtering effect of the ground glass diffuser. To further explain the underlined mechanism that the OAM-dependent speckles can work for the OAM mode sorting, a proof-of-principle experiment is provided in Supplement 1 Note 6, illustrating the evolution features of the cross-correlation map among OAM-dependent speckles corresponding to PVBs with topological charges ranging from $\alpha = - {3}$ to $\alpha = {3}$. It is worth acknowledging that the cross-correlation annulus cannot give the true value of the topological charge. However, since its cross-correlation annulus is only determined by the absolute value of the topological charge difference, it is sufficient to sort the OAM modes especially when the detecting modes are well-defined and the reference OAM-dependent speckle is known as prior information. In contrast to traditional OAM spectrum measurement approaches, the cross correlation of the OAM-dependent speckles greatly reduces the system complexity in terms of the amount of required optical components (just a piece of ground glass diffuser here) and stability (no need for strict interferometric alignment). The calibration only necessitates a prior OAM-dependent speckle, which is easier to implement and much more stable compared with the complex field TM measurement and the training process in the deep-learning approach.
Fig. 5. Experimental analysis of the OAM-dependent speckles. (a) Measured cross-correlation maps between the reference OAM-dependent speckle (${{ I}_{- 8}}$) and the other ones (${{ I}_{- 8}}$, ${{ I}_{- 5}}$, ${{ I}_{- 2}}$, ${{ I}_2}$, ${{ I}_5}$, and ${{ I}_8}$). (b) Normalized central cross-correlation curves corresponding to the cross-correlation maps (${ Y}$ shift direction) in (a) as a function of $\theta$. (c) Measured orthogonal relations between the OAM-dependent speckles corresponding to PVBs with topological charges ranging from $\alpha = - {20}$ to $\alpha = {20}$ stepped by 1. (d) OAM-dependent speckle decorrelation bandwidth.
Fig. 6. 8-bit grayscale OAM-encoded data, which is demultiplexed by the OAM-dependent speckle basis. (a) 8-OAM bases (theoretical) and the corresponding OAM-dependent speckles measured experimentally. (b) Theoretical profiles of the multiplexed intensity and phase corresponding to a grayscale intensity value of 86. (c) Experimentally measured speckle pattern obtained by transmitting the multiplexed data through a ground glass diffuser. (d) Experimental OAM spectrum of intensity value 86 demultiplexed by the OAM-dependent speckle basis. Statistical analysis of the experimentally calculated OAM spectra as a function of (e) gray scale pixel value and (f) bit number. (g) Reconstructed intensity value versus the ground truth intensity value.
The orthogonal relation among the OAM-dependent speckles corresponding to PVBs with topological charges ranging from $\alpha = - {20}$ to $\alpha = {20}$ is also experimentally verified by the diagonal correlation feature, as indicated in Fig. 5(c). It is also shown that the correlation coefficient between the neighboring integer orders is around 0.25, which can be regarded as decorrelated speckles. To give a more quantitative characterization of the decorrelation bandwidth (defined by the change in topological charge to reduce the correlation coefficient by half), the correlation coefficients of the OAM-dependent speckles corresponding to PVBs with topological charge centered at $\alpha = {4}$, 7, and 10 stepped by 0.1 are provided in Fig. 5(d). The OAM-dependent speckle decorrelation bandwidth is found to be ${\sim}{0.5}\;{\rm nm}$. Therefore, for a discrete decorrelated random speckle basis, an integer order based topological charge separation is enough for the decorrelation of the OAM-dependent speckles and guarantees efficient demultiplexing.
D. Demultiplexing of OAM-Encoded Data Using Scattering Medium
For demultiplexing of OAM spatial mode, OAM-encoded data from the super-pixel wavefront shaping is generated and the OAM-dependent speckle basis is employed as the key. It is worth noting that the demultiplexing only corresponds to the receiver part in a typical communication system. A piece of ground glass is employed to generate the OAM-dependent speckles and enable the measurement of the OAM spectrum with the assistance of a camera. The ground glass does not correspond to the atmospheric turbulence along the transmission in free-space communication. For an 8-bit grayscale OAM-encoded data, the OAM basis composed of eight PVBs (i.e., $\alpha = \pm {4}$, $\pm {3}$, $\pm {2}$, and $\pm {1}$) is employed for the data encoding, as defined in Fig. 6(a). For each PVB basis, the OAM-dependent speckle, i.e., the OAM-dependent transmission function $H(x,\;y,\;\alpha)$ shown in the second row in Fig. 6(a), is correspondingly measured. The 8-bit grayscale intensity value, i.e., $O(\alpha)$, is then encoded, representing 256 gray levels. For example, the grayscale intensity value 86 is transformed into the binary byte of “01010110” with four 1-value bits (i.e., ${N} = {4}$). Figure 6(b) gives the theoretical intensity and phase profiles of the encoded beam. Taking advantage of the decorrelation characteristics of the ground glass diffuser, the measured speckle pattern $I(x,\;y)$ [e.g., Fig. 6(c)] obtained after the encoded signal transmitting through the ground glass diffuser can be written as
Therefore, the encoded signal $O(\alpha)$ can be demultiplexed through a pseudoinverse operation (the Moore–Penrose inverse), and the reconstructed OAM spectrum is shown in Fig. 6(d). Here, the reconstructed power ratio of each OAM channel is normalized by the maximum value of the original calculated ones to promote the available threshold range for binarization. From the calculated OAM spectrum in Fig. 6(d), it is clear that the correct binarization can be realized by setting the threshold between 0.009 and 0.687. To further investigate the statistical characteristics of the OAM-dependent speckle basis for data demultiplexing, grayscale intensity values from 1 to 255 are considered. In this case, the binarization should be realized simultaneously for all the data by setting the same threshold. Therefore, all the calculated OAM spectra should be displayed together. Here, the normalized power ratio of the calculated OAM spectra as a function of intensity values and bit numbers is shown in Figs. 6(e) and 6(f), respectively. It is clear that for both results the threshold can be set in a very broad range between 0.147 to 0.545 for efficient binarization. By setting any threshold values within this range, the reconstructed intensity values versus the ground truth intensity values are obtained in Fig. 6(g). The smooth and linear profile verifies the 100% correct intensity value reconstruction demultiplexed by the OAM-dependent speckle basis.
Fig. 7. Results of the 8-bit grayscale OAM-encoded data. (a1) Sent and (a2) received grayscale images (Peppers, ${100} \times {100}\;{\rm pixels}$). (a3) Statistical analysis of OAM spectra as a function of pixel number. (b1) Sent and received grayscale images (Peppers, ${200} \times {200}\;{\rm pixels}$) with (b2) single or (b3) partitioned basis. (c) Error rates of the results in (a2), (b2), and (b3).
Leveraging the above OAM encoding and decoding strategy, an 8-bit grayscale image (Peppers [76], $m \times n$ pixels) is used for data demultiplexing, as shown in Fig. 7. The image is sent and decoded pixel by pixel, which can be written as
where ${I_{xy\: \times \:mn}}$ is reshaped from $mn$ speckle patterns (each of which is $x \times y$ in size) corresponding to the image size of $m \times n$, ${H_{xy\: \times \:\alpha}}$ is reshaped from $\alpha$ OAM-dependent speckle basis, i.e., $\alpha = {8}$ for 8-bit grayscale data, and ${O_{\alpha \: \times \:mn}}$ is the target OAM spectrum to be reconstructed. Here, a ${100} \times {100}$ pixels sized image [Fig. 7(a1)] is first considered. Statistical analysis of the OAM spectra as a function of the pixel number is displayed in Fig. 7(a). It is worth noting that the OAM-dependent speckle basis ${H_{xy\: \times \:\alpha}}$ is measured in the middle when sending the ${100} \times {100}$ speckle images to suppress the decorrelation effect of the OAM-dependent speckles. It is found that the clear band gap of the threshold selection range may become narrowed for the speckle patterns measured with a larger time interval with respect to the OAM-dependent speckle basis. The scatters within the band gap region would lead to inaccurate demultiplexing. Notwithstanding, high-quality image reconstruction is realized in Fig. 7(a2) after taking the binarization operation. The error rate (to quantitatively characterize the effectiveness of demultiplexing using scattering medium) of this 8-bit grayscale OAM-encoded data is calculated to be 38.24 dB with only one set of OAM-dependent speckle bases, as shown in Fig. 7(c).Fig. 8. Results of the 24-bit RGB OAM-encoded data. (a) Schematic diagram of the 24-bit RGB encoding scheme using 24-OAM multiplexing. (b1) Sent and (b2) received RGB color images (Peppers, ${100} \times {100}\;{\rm pixels}$). (c) Error rates in each RGB channel.
To verify the capability of data demultiplexing in a larger size, the image of interest is increased to ${200} \times {200}$ pixels as shown in Fig. 7(b1). Here, two decoding strategies are employed respectively; one is using a single set of OAM-dependent speckle bases for the whole speckle images, and the other one is dividing the ${200} \times {200}$ image into five groups, using a partitioned basis to reconstruct each part and combining the reconstructed parts together as a whole, as shown in Figs. 7(b2) and 7(b3). We can see an obvious decline in the error rate for the first single basis strategy, i.e., 33.23 dB, and the error rate value returns to 36.66 dB for the partitioned basis, as shown in Fig. 7(c). Both reconstructed images show high fidelity in contrast with the ground truth image. To further investigate the stability of the system that is closely related to the demultiplexing efficiency, the speckle decorrelation effect is evaluated in detail in Supplement 1 Note 7.
Finally, the concept of OAM-dependent speckles in the data demultiplexing is further extended to a color image that is weighted by three primary colors, i.e., red, green, and blue. The encoding scheme of the 24-bit RGB data is provided in Fig. 8(a). The PVBs with topological charges ranging from ${-}{12}$ to 12 are considered as the basis, where the red color takes the topological charges from ${-}{12}$ to ${-}{5}$, the green one from ${-}{4}$ to 4, and the blue one from 5 to 12, respectively. Therefore, each primary color layer is composed of an 8-bit grayscale image and the mixture of the three layers yields the RGB color image. Figure 8(a) shows the multiplexed intensity and phase profiles of a specific color, i.e., red, 231; green, 137; blue, 113. It is worth noting that the considered image is ${100} \times {100}$ and only one set of OAM-dependent speckle bases is employed. The reconstructed RGB color image in Fig. 8(b2) matches well with the ground truth image in Fig. 8(b1), except for a small number of visible noise points. The error rates in each of the RGB channels, i.e., 33.23 dB for red, 33.98 dB for green, and 33.72 dB for blue as shown in Fig. 8(c), also reflect the high demultiplexing efficiency using the OAM-dependent speckle basis even it is extended to a 24-bit RGB OAM-encoded data.
E. Discussion
Compared with traditional methods for the OAM mode sorting, the cross-correlation approach takes full advantage of the scattering effect to achieve OAM-dependent speckles and requires only a piece of ground glass as the key element. Besides, the cross-correlation calculations are based on the speckle patterns, which are incoherently measured after the vortex beam transmits through the ground glass diffuser, without using any sophisticated optical components (such as a spiral phase plate, fork hologram, or $q$-plate) or complex optical alignment (such as in cascaded interferometers). With prior information on a specific OAM-dependent speckle pattern for calibration, the OAM mode sorting can be realized by characterizing the radius of the cross-correlation annulus. No further characterization of the scattering diffuser is needed, such as tailoring the scattering effect actively by the wavefront shaping or the coherently calibrated TM using the interferometric technique. In addition, the cross-correlation operation of the speckle patterns is very simple and highly efficient, thus contributing to their wide uses in speckle-correlated imaging and speckle image velocimetry. The partially digital cross-correlation operation of the OAM-dependent speckles in this experiment can be readily converted into a purely optical demonstration (e.g., using a Mach–Zehnder interferometer) in future work, which intends to create a new area for both correlation analysis and OAM mode characterization. In addition, we would like to stress that the proposed technique can also be extended to sort other vortex modes (e.g., LG modes) with both the azimuthal and radial indices identified.
More importantly, the OAM-dependent speckles corresponding to the vortex beam with different topological charges are used as a set of OAM-dependent speckle bases for the demultiplexing of OAM-encoded data. Distinct from commonly used approaches where the TM of the scattering diffuser is calibrated, or the speckles are thoroughly analyzed with the assistance of the deep-learning method, the demultiplexing technique proposed here only requires the calibration of the OAM-dependent speckle basis [i.e., 8(24) speckle patterns for 8(24)-bit OAM-encoded data]. The latter is much simpler than the calibration of a traditional TM or the training process in a deep-learning approach. For a ${100} \times {100}$ sized image, the required measurement of the OAM-dependent speckle basis for the 8(24)-bit OAM-encoded data only accounts for 0.08% (0.24%), which takes up a relatively small portion of the whole data volume. A ground glass diffuser is the only key element that is required, greatly reducing the demultiplexing complexity for the OAM-encoded data. Even compared with the classical OAM demultiplexing techniques such as phase flattening, the OAM-dependent speckle based approach is still attractive with promising applications due to its superior advantages of low cost, easy implementation, and excellent robustness. For demultiplexing of whatever OAM-encoded data (e.g., 8-bit gray or 24-bit RGB data), we only need a ground glass diffuser, a camera, and the prior information of the OAM-dependent speckle bases that can be easily obtained in the experiment. Because of the easy-to-implement approach of measuring the single set of speckle bases, we do not need to worry about a significant shift of the ground glass that reduces the correlation and thus deteriorates the error rate.
It is also worth noting that the demultiplexing rate of OAM-encoded data using the OAM-dependent speckle basis here is mainly restricted by the frame rate (i.e., 15 fps here) of the CCD camera. However, this is only a technical bottleneck, not the intrinsic limitation stemming from the concept of OAM-dependent speckles for spatial mode sorting and demultiplexing. It is speculated that by using a high-speed camera or other high-speed detecting techniques, the OAM-encoded data acquisition rate can be greatly accelerated. In addition, the speckle decorrelation effect can be further suppressed to reduce the error rate by fixing the ground glass diffuser and the camera together to reduce the relative shift from environmental vibrations, especially when a much larger data volume is sent and demultiplexed.
An entirely new approach to sorting and demultiplexing the spatial modes is proposed and demonstrated, with a complete theoretical calculation and a proof-of-principle experiment. The results can further advance the study of vortex photon behaviors in strongly scattering systems, which constitutes a good complement to the traditional research scope of vortex optical fields. On the other hand, the linear mapping relation between the input and output optical fields in the scattering media is strategically harnessed by introducing the orthogonal vortex modes, which accelerates the research and development of effective information extraction from speckles. Therefore, we hope this work can advance the research of OAM-dependent speckle correlation in scattering systems and add new knowledge to the properties of vortex modes and their potential applications as well.
3. CONCLUSION
In conclusion, the characteristics of OAM-dependent speckles, originating from the vortex beam transmitting through a ground glass diffuser, are theoretically and experimentally investigated. It is found that, by imposing a cross correlation between the OAM-dependent speckles, an annulus can be obtained on the cross-correlation map with its radius dependent on their topological charge difference. Therefore, the OAM modes can be effectively sorted with prior information on a specific OAM-dependent speckle as a reference. Furthermore, an OAM-dependent speckle basis can also be utilized as a competitive candidate for the demultiplexing of OAM-encoded data. The comparatively low error rates of both 8-bit grayscale and 24-bit RGB OAM-encoded data indicate the feasibility of the OAM-dependent speckle-based demultiplexing. This work reveals the fundamental features of OAM-dependent speckles and paves an appealing method for OAM-encoded data demultiplexing in practical applications.
Funding
Guangdong Basic and Applied Basic Research Foundation (2020A1515111143); Natural Science Foundation of Guangdong Province (2021A1515011532, 2023ZDZX3022, 2024A1515012152); Shenzhen Government’s Plan of Science and Technology (JCYJ20220818100019040, JCYJ20230808105713028, RCYX20210609103157071).
Acknowledgment
R. M. conceived the idea and wrote the original manuscript. R. M. and K. H. L. conducted the experiments. R. M. and Z. W. prepared the numerical calculation. K. H. L. and Z. W. realized the software control. O. K. and S. P. established the theoretical model for intensity correlations. R. M. and J. L. funded the work. J. L. and J. S. H. supervised the project. W. L. Z., D. Y. F., and A. S. L. G. discussed the results. All the authors discussed the results and contributed to the writing of the manuscript.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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