The estimation of the state of polarization (SOP) is critical in various fields, such as optical communications,1 radar imaging,2 and solar astronomy.3 Polarimeters, instruments designed to measure the SOP of light, have received significant attention. Formerly, the focus was primarily on the linearly polarized light and was concentrated on measuring the initial three components of the Stokes vector.4^,5 During this period, information about the circularly polarized state, associated with the fourth component, was not obtainable.6^,7 Subsequently, comprehensive polarimeters capable of measuring the full Stokes vector were investigated. However, conventional SOP measuring methods, such as the rotating quarter-wave plate technique, necessitate the employment of optical elements like polarizers. The inclusion of diverse optical components increases the complexity of the system, and the measurement results are sensitive to environmental noise. Therefore, it is necessary to explore a simple and effective SOP measurement. Apart from traditional methods, metasurfaces, engineered with designed micronanostructures, have been utilized to conduct SOP measurements with enhanced system compactness.8^–10 Nonetheless, they present challenges in design and manufacturing, coupled with limited adjustability and reconfigurability. Furthermore, fibers have been employed to detect SOP, due to their inherent polarization sensitivity.11 Based on the theory of multicore polarization interference,12 researchers have utilized three-core photonic crystal fibers for SOP measurement. However, this approach can only detect left- and right-handed polarized lights with varying elliptical polarizations. Moreover, the utilization of photonic crystal fibers results in considerable expense. Therefore, there is an urgent need to develop a compact and cost-effective system for measuring the Stokes parameters of light.

Considering easy implementation and cost reduction, the multimode fiber (MMF) emerges as an economically practical and lightweight alternative.13 The large core diameter of the MMF enables the excitation of multiple modes with different phase velocities, and these applications are based on the principle of mode coupling. The coupling of the different modes leads to speckle patterns with light and dark features at the output. The speckle patterns contain both amplitude distribution and phase information for all modes.14^–16 Changes in the external environment or input light induce the relative phase between modes and affect the speckle. Thus it is feasible to establish a correlation between the measurement to be conducted and the speckle. The MMF has been employed for the sensors of external factors,17 such as temperature,13^,18 curvature,19^–21 reflective index,16 and magnetic field.22 It also has applications in endoscopy and the transmission of complex images,14^,23^,24 and has also proven to be an effective medium for measuring the properties of incident light.25^,26 The above application can be achieved by solving the transmission matrix of an MMF to establish the mapping of inputs and outputs.27 However, gradual changes in the external environment can impact the transmission matrix, leading to an accumulation of errors during the transmission process.28 To keep the system stable, researchers have protected MMFs from external interference or developed interference-insensitive MMFs. Others have applied a real-time recalibration of the transmission matrix of the MMF.29 In addition, researchers employ neural networks to establish mapping relationships between speckles and the parameters to be measured in order to enhance the robustness of measurement systems based on MMF speckle.25^,30 We have verified that the tolerance of MMF to environmental noise can be improved by a convolutional neural network (CNN) in our previous experiments.31 As a result, measurement of input light SOP can be achieved by monitoring and analyzing the speckle patterns generated by the MMF.

In this paper, we demonstrate an MMF speckle Stokes polarimeter. Multiple MMF speckles were obtained by changing input SOPs. We employed a CNN to efficiently establish a correlation between SOP and speckle intensity distribution. The system does not require optical alignment and laborious microstructure design. The MMF mitigates system complexity and cost and enables applicability for different wavelengths. The CNN excludes environmental impacts, thereby enhancing efficiency and flexibility. For applications related to polarization, the proposed method has significant value.

The distribution of light intensity emitted from the MMF can be explained by Eq. (1), which is a result of different modes of coupling.32 In Eq. (1), the amplitude and complex phase of the $m$th mode are represented by $a_m$ and ${\varphi }_m$. $M$ denotes the number of modes. Interference occurs on the end facet of an MMF between different modes. If the phase delay between modes is $2\pi$ (or a multiple of that), the interference is constructive. If it is $\pi$, the interference is destructive. Other possible phase delays produce partially destructive or partially constructive intermediate interferences, ultimately forming a bright and dark scattering pattern at the other end of the MMF.33 Change in the input SOP will alter the phase difference between the $x$- and $y$-polarization quantities, changing the complex phase of modes, thereby affecting the intensity distribution, $$I(x,y)=|A(x,y){|}^2=\sum _{m=1}^M\sum _{n=1}^M{a}_m{a}_n\exp [j({\varphi }_m-{\varphi }_n)].$$(1)

After obtaining speckles related to SOPs from MMF,26 a CNN is employed to establish a mapping relationship between them. The reason for using a CNN is its wide range of applications in image classification and segmentation.34^,35 Research has demonstrated the effectiveness of CNN in complex nonlinear modeling and classification tasks for detecting intensity within MMF transmission systems.30^,36 Furthermore, this approach eliminates the requirement for additional reference structures and solves the problems associated with minor temperature drift in MMF transmission.14 Therefore, the subsequent task is to denote the SOPs as CNN labels, $$S_0^2=S_1^2+S_2^2+S_3^2.$$(2)

Most optical instruments have difficulty in the measurement of high-frequency complex-valued electric and magnetic fields directly.32 To accurately and comprehensively describe the polarization state of light in measurable quantities, Stokes introduced the Stokes parameters (or Stokes vectors) $[S_0,S_1,S_2,S_3]$.37 The vector contains four optically measurable quantities and they satisfy the relationship shown in Eq. (2). On the basis of Eq. (2), Henri Poincaré proposed the Poincaré sphere for a clear and graphical representation of SOPs. Each SOP corresponds to a point on the surface of the sphere.38 An SOP with known Stokes parameters $[S_0,S_1,S_2,S_3]$ can be considered as a point on a sphere of radius $S_0$ with coordinates $(S_1,S_2,S_3)$, or as a vector that points from the origin $(\mathrm{0,0},0)$ to the point $(S_1,S_2,S_3)$, as depicted in Fig. 1(a). For polarized light, $S_0$ equals 1.

During the measurement process, there is a natural discrepancy between the predicted and actual vectors, as depicted in Fig. 1(b). The root-mean-square error (RMSE) is often used as a criterion for the estimation accuracy of this discrepancy.39 Thus it can act as the loss function of the CNN. Though $S_0$ is fixed, it is still involved in network learning to constrain the other three Stokes parameters. The RMSE of $S_0$ theoretically converges to 0. So the physical meaning of the entire RMSE is approximated as the length of the difference between the two Stokes vectors, as shown in Fig. 1(b), which is utilized to evaluate the accuracy of measurements.

The experimental setup is shown in Fig. 2. The wavelength of the laser is 1550 nm. MMF (SI 105/125-22/250, YOFC) has a core diameter of $105\mu \mathrm{m}$ and a length of 10 m. In the measurement process, it is necessary to modulate the light of a known fixed SOP to achieve an output of an arbitrary SOP to be measured. Therefore, the Mueller matrix is introduced as a $4\times 4$ order matrix describing the matrix of the change in SOP during transmission and its Stokes parameters are calculated mathematically. Multiplying the Stokes parameter of the incident light and the Mueller matrix of the transmission medium gives the Stokes parameter of the outgoing light after passing through the transmission medium,4^,5 as shown in Eq. (3). The programmable motorized polarization controller (MPC) consisting of two autonomous quarter-wave plates is utilized to achieve arbitrarily polarized outputs, as shown in Fig. 3(a). The plates possess the ability to rotate between 0 and 225 deg, with a precision of 0.225 deg per step. Later in the text, the degree of rotation will be expressed based on the step of rotation. These dual quarter-wave plates, which can be adjusted independently by software, enable one to produce any SOP output to be used as the input of MMF. The CCD (SP620U-1550, Spiricon, image resolution $1200\times 1600$) captures the intensity distribution of the speckle patterns corresponding to different SOPs from MMF, as presented in Fig. 3(b). The $1\times 4$ matrix below each speckle is the corresponding Stokes parameters, $$S^{\prime }=MS=\left[\begin{array}{c}{S}_{0^{\prime }}\\ S_{1^{\prime }}\\ S_{2^{\prime }}\\ S_{3^{\prime }}\end{array}\right]=\left[\begin{array}{cccc}{m}_{00}& m_{01}& m_{02}& m_{03}\\ m_{10}& m_{11}& m_{12}& m_{13}\\ m_{20}& m_{21}& m_{22}& m_{23}\\ m_{30}& m_{31}& m_{32}& m_{33}\end{array}\right]\left[\begin{array}{c}{S}_0\\ S_1\\ S_2\\ S_3\end{array}\right].$$(3)

The image size of the speckle after cropping is $600\times 600$. To minimize loading time, all data have been preprocessed. A $3\times 3$ kernel is employed to perform maximum pooling on the region, resulting in a $200\times 200$ input for the CNN. Through this process, pertinent information is encompassed, and the loading time for the CNN is reduced. The CNN architecture employed is illustrated in Fig. 4, which consists of several residual blocks, convolution, and pooling layers. To better extract detailed information, reduce the number of parameters, and avoid overfitting, the convolutional layers do not change the length or width of the input; it only changes the number of channels, and the dimensionality is reduced mainly through the pooling of the layers. The final fully connected layer generates an estimated value of Stokes parameters. The learning rate, batch size, and epoch are set to 0.001, 128, and 100.

Investigating the capability of the system required an SOP data set covering the full Poincaré sphere. We regulated the SOP through MPC adjustment with a rotational angle of 4.5 deg (20 steps). Two quarter-wave plates rotated from 0 to 225 deg individually and 2600 images were collected as a group, resulting in an SOP distribution traversing the entire Poincaré sphere. It should be noted that although the rotation of the MPC increases linearly, its impact on the Mueller matrix is nonlinear. Therefore, some SOPs in the data set were sparse, while others were dense. The data were subjected to a filtering process with the objective of achieving a more uniform distribution across the sphere, which is presented in Fig. 5(a). To eliminate environmental impact, 16 groups of SOPs that covered the entire Poincaré sphere were collected. The data set was divided into a training set, validation set, and test set at an 8:1:1 ratio. In order to evaluate the ability of CNN to predict unknown SOPs, there is no overlap between SOPs in the sets mentioned above. Following the training process, the network generated an RMSE value of 0.0602, which represents the magnitude of the Stokes vector difference between the predicted and actual SOP vectors. For clarity, a distribution function of the loss in the test data set is presented in Fig. 5(b), whereas the RMSE of each point is shown in three-dimensional form in Fig. 5(c). RMSEs of the four Stokes parameters in the test data set are 0.0189, 0.0783, 0.1076, and 0.0694. It should be noted that although $S_0$ is a fixed value, it is still included in the label and used in the training process due to the relationship between the Stokes parameters, as shown in Eq. (2). The inclusion of $S_0$ as a physical constraint in the learning process reduces the RMSE and facilitates network convergence. For clear visualization of the precision of the Stokes parameters prediction, we randomly selected 30 SOPs from the test set and illustrated a comparative graph of their predicted and actual Stokes parameters values ($S_1$, $S_2$, and $S_3$) in Fig. 5(d). The outcome demonstrates the capability of the system to measure the four Stokes parameters.

To investigate the influence of polarization range on prediction accuracy, we extracted a partial data set from the global data set, consisting of $20\times 20$ SOPs. (Two quarter-wave plates rotated from 0 to 90 deg, individually.) The RMSE of the partial data set is 0.0658, which is similar to the global data set’s RMSE of 0.0602. Both RMSEs are smaller than the vector differences between adjacent SOPs (0.2015). The slight difference of RMSEs between full and partial data sets is caused by the edge SOPs of the partial sphere data set. Figure 6(a) shows the partial SOP distribution. The distribution of partial test data set RMSE is shown in Fig. 6(b). It demonstrates that the measurement accuracy of the system is not related to the range of the SOP. Figures 6(c) and 6(d) depict the RMSE distribution on the Poincaré sphere and predictions corresponding to each Stokes parameter ($S_1$, $S_2$, and $S_3$), respectively.

To improve the estimation accuracy, we reduced the MPC rotation interval to generate denser-distribution data sets of SOPs. The MPC rotation increments are set at 1, 5, 10, and 20 (0.225, 1.125, 2.250, and 4.500 deg). We have demonstrated the accuracy similarity with a step size smaller than 20. Therefore, for each interval, 10 sets of $20\times 20$ SOP speckles were collected (Two quarter-wave plates rotate from 0 deg to 20 times of interval individually as a set). Figure 7 illustrates the distribution of SOPs on the Poincaré sphere with different rotation steps. The density and homogeneity of the SOPs increase as the interval reduces with the same number of SOPs. Additionally, the speckle changes between adjacent SOPs are not distinguishable to human eyes.

Table 1 shows the outcomes for each network. By reducing the MPC rotational angle, the interval between adjacent SOP vectors is diminished, resulting in a more uniform and dense SOP distribution, thus enhancing the measurement precision. The RMSEs are smaller than the corresponding vector difference. Figure 8 depicts the RMSE distribution functions of test data sets, whereas Fig. 9 presents the loss distribution in 3D structure for test data sets with diverse MPC rotation steps. The concentration of the loss distribution increases, and the mean value decreases as the MPC rotation angle decreases, indicating accuracy improvement through training data optimization. Furthermore, as the data set distribution becomes denser and the distribution of neighboring SOPs becomes more linear, the likelihood of sudden large RMSEs decreases, increasing the reliability of measurement. Figure 10 compares the Stokes parameters of 30 selected SOPs in the test data sets, further illustrating that using data with smaller vector difference intervals increases accuracy. The results obtained in experiments were considerably lower than the stated accuracy. It should be noted that with an MPC rotation step size of 1 (0.225 deg), the variations in vector values among different SOPs are minor, and the RMSE is 0.0042. The RMSEs for the four Stokes parameters are 0.0021, 0.0043, 0.0050, and 0.0028, correspondingly. The system measures the four Stokes parameters with similar accuracy and is well-balanced in its sensitivity to each value. This limitation in accuracy is due to the minimum rotation angle of the quarter-wave plate in the MPC. It reveals that the system is capable of achieving highly precise SOP measurements.

Wavelength influences the intensity distribution of speckles.25 As depicted in Fig. 11(a), the speckles have different intensity distributions with the same SOP at different wavelengths. Therefore, it is imperative to assess the system for its applicability to different wavelengths. Since the measurement principle of the system is similar at different wavelengths and the resolution is mainly related to SOP distribution density, a 10 nm spectral band is used to verify wavelength applicability, and the MPC rotation increment was set at 80 (18 deg). The laser emits light at a single wavelength, ranging from 1545 to 1555 nm in increments of 0.5 nm. And $10\times 10$ SOP speckles were collected as a set, as shown in Fig. 11(b). (Two quarter-wave plates rotate from 0 to 180 deg individually as a set). We collected five sets of SOP speckles, covering the entire Poincaré sphere for each wavelength. The RMSEs for different wavelengths are depicted in Fig. 11(c). The minimum RMSE is 0.1678 at 1555 nm, while the maximum is 0.3400 at 1550 nm. Given that the SOPs in the data set are sparse, these RMSEs are smaller than the interval between adjacent SOPs (0.7415). The results indicate that the largest RMSE occurs at 1550 nm, which is the same wavelength used in the previous experiment. Therefore, using other wavelengths in the system can achieve similar SOP measurement accuracy with similar SOP distribution density, making it applicable to different wavelengths with interest.

In this paper, we proposed and demonstrated an MMF speckle Stokes polarimeter. The MMF is employed to generate specific speckle patterns with specified SOPs, while a CNN is utilized to establish a mapping relationship between them. We demonstrate the polarized optical Stokes parametric measurement capability of the system. The system can achieve a minimum RMSE of 0.0042. The RMSEs for the Stokes parameters in this case are 0.0021, 0.0043, 0.0050, and 0.0028, respectively. The system currently performs highly precise single-point polarized optical Stokes parameter measurements at different wavelengths. It enables a low-cost and straightforward Stokes polarimeter that provides great value for polarization-related applications. Also, it has the potential to be used for full Stokes parameter and polarization measurements of complex optical fields in the future.

Yuxuan Xiong received her BS degree from Huazhong University of Science and Technology (HUST), Wuhan, China, in 2021. She is currently working toward her PhD at the School of Optical and Electronic Information, HUST, Wuhan, China. Her current research interests include multimode fiber, fiber sensors, and optoelectronic systems.

Ting Jiang received her BEng degree in telecommunications engineering from Beijing University of Posts and Telecommunications (BUPT), Beijing, China, in 2020. Since 2020, she has been working toward her PhD at the School of Optical and Electronic Information, HUST, Wuhan, China. Her research interests include optical performance monitoring, machine learning, and digital signal processing.

Hao Wu received his BS, MS, and PhD degrees from HUST, Wuhan, China, in 2013, 2016, and 2019, respectively. His postdoctoral research at HUST was focused on the machine learning algorithms for distributed optical fiber sensing. Since 2024, he has been a research associate at HUST. His current research interests are the integration of artificial intelligence and optical fiber.

Zheng Gao received his BEng degree from the School of Optical and Electronic Information, HUST, Wuhan, China. Since 2020, he has been working toward his PhD at the School of Optical and Electronic Information, HUST, Wuhan, China. His current research interests include multimode fiber and fiber optics sensing.

Shaojun Zhou received his BS degree from Hubei University of Technology, Wuhan, China, in 2021 and his MD degree from HUST, Wuhan, China, in 2024.

Zhao Ge received his BS degree from Jianghan University, Wuhan, China, in 2019 and his MD degree from HUST, Wuhan, China, in 2022. He is currently working toward a PhD at the School of Optical and Electronic Information, HUST, Wuhan, China.

Siqi Yan received his BS and PhD degrees in optics engineering from HUST in 2013 and 2018, respectively. From 2019 to 2021, he worked as a postdoctoral researcher in the Silicon Photonics for Optical Communications Center at the Technical University of Denmark. In November 2021, he joined the HUST as an associate professor. His primary research interests are high-performance silicon/graphene integrated optoelectronics chips and their applications in optical communication systems.

Ming Tang received his BEng degree from HUST in 2001 and his PhD from Nanyang Technological University, Singapore, in 2005. His postdoctoral research at the Network Technology Research Centre was focused on the optical fiber amplifier. From 2009, he was a research scientist in the Tera-Photonics Group, RIKEN, Japan. Since 2011, he has been a professor at HUST. His current research interests include optical fiber-based linear and nonlinear effects for communication and sensing applications.