The lightwave possesses several fundamental physical properties including intensity, phase, spectrum, and polarization. The multi-dimensional measurement of the lightwave finds many applications across different fields, such as fiber optic communications^[1] and monitoring^[2], chemical composition analysis^[3], material surface texture detection^[4,5], and biosensing^[6].

Conventional polarimeters and spectrometers often have considerable bulkiness and weight^[7,8], failing to meet the requirements for miniaturization in practical applications. To solve these problems, some metasurface based spectrum analyzer and polarization analysis system designs have been proposed in recent years^[9–13], in which the spectrum or polarization information is mapped into the spatial domain by interacting with the metasurface. At the same time, with the rapid development of complex lightwave field manipulation technologies, the corresponding demand for multi-dimensional measurement of lightwaves has received increased attention.

The simultaneous measurement of the spectrum and polarization of lightwaves can be realized by multiplexing techniques^[14,15], while most systems hold increased footprint and complexity. Recently, some novel miniaturization designs have achieved simultaneous measurement through direct integration of metasurface and detector arrays^[16–20], which require precise design and fabrication with limited measurement dimensions. The plasmonic metasurface scheme shows a polarization measurement mean squared error (MSE) of more than 1×10−2 and a spectral resolution of 15 nm^[19], and the nematic liquid crystal scheme has a polarization measurement MSE of 5×10−3 and a spectral resolution of 7 nm^[20]. However, these schemes demonstrate limited performance characterized by large reconstruction errors of Stokes parameters and low measurement resolution, and the simultaneous multi-dimensional reconstruction of lightwaves with high spectral resolution and low polarization measurement error remains challenging.

In this paper, we propose a novel single-shot all-fiber high-resolution spectropolarimeter, achieving simultaneous measurement of the spectrum and polarization of incident lightwaves by measuring the speckle patterns generated from the end of a multi-mode fiber (MMF). The experimental results show that the proposed system achieves a spectral resolution of 100 pm and a polarization resolution of 0.001437. The polarization measurement errors for three Stokes parameters are 3.37%, 1.01%, and 0.84%, respectively, and the overall MSE is 5.3×10−5, which is two orders higher than the state-of-the-art results of spectropolarimeters^[20,21]. The spectropolarimeter design based on MMF shows low loss and generalization capability, enabling it to measure multiple dimensions of lightwaves with high spectral resolution and low polarization measurement error.

Assuming all the guided modes that can be supported in MMF are excited, the resulting electric field distribution at the end of the MMF can be expressed as^[22]E(r,θ,λ,p,x)=∑mAmΦm(r,θ,λ,p)exp{−i[βm(λ,p)x−ωt+φm]},(1)where Am represents the amplitude of the mth mode, φm denotes the initial phase, βm is the propagation constant, and Φm is the spatial profile. Variations in both spectrum and polarization introduce changes in the phase delays βm(λ)x, leading to alterations in the speckle patterns^[23].

Figure 1(a) shows the speckle patterns corresponding to various wavelengths and polarization states, respectively, demonstrating a complex yet deterministic mapping relationship. This fixed mapping can be used for the determination of the spectrum and polarization by the reconstruction method. The transmission matrix algorithm is a common reconstruction method for the measurement of the spectrum and polarization. Before measurement, the transmission matrix algorithm has a calibration process to acquire a transmission matrix D composed of various speckles of different spectra and polarizations. The spectrum S and polarization P of the lightwave can be reconstructed from the intensity distribution I of speckle by [Sm,P3n]T=D(m+3n)×k−1Ik×1.

The experimental setup is shown in Fig. 1(b). The lightwave under test (LUT) is generated by a tunable laser and an electro-polarization controller (EPC). The tunable laser generates the lightwave at the required wavelength, and the EPC adjusts the polarization state. After passing through a 90:10 optical coupler, 90% LUT is coupled into MMF by a fixed ferrule connector/physical contact (FC/PC) connector, and the resulting speckle pattern generated at the end of the MMF is imaged onto a charge-coupled device (CCD) camera (Xenics Bobcat-320-Star) via a lens for measurement. The remaining LUT is divided evenly into two beams via a 50:50 optical coupler. These two beams are separately directed into an optical spectrum analyzer (OSA) and a polarization analysis system (PAS) for reference. Similar to the design of rotating waveplate polarimeters, PAS (POD-201, General Photonics) comprises piezoelectric actuator-driven fiber squeezers oriented 45° from each other. The fiber squeezer is driven by voltage signals, inducing linear birefringence in the fiber upon compression. An optical spectrum analyzer (Yokogawa, AQ6370C) is used to obtain the spectral information. The camera offers 320 × 256 image resolution at a 30 Hz frame rate, and the pixel pitch is 20 µm.

To quantify the impact of the spectrum on this sensitivity, the spectral correlation function of the speckle intensity^[22] is calculated for estimating the spectral resolution: C(Δλ,x)=⟨I(λ,x)I(λ+Δλ,x)⟩/[⟨I(λ,x)⟩⟨I(λ+Δλ,x)⟩]−1,(2)where the term ⟨…⟩ represents the average over all wavelengths, denoted by I(λ) for the intensity of the speckle at the specific wavelength λ. In the system, we adopt a commercial standard step-index MMF with a length of 2 m [core diameter (CD) = 105 µm, numerical aperture (NA) = 0.22]. The spectral correlation function is shown in Fig. 2(a). The wavelength shift for the correlation to decrease by half is 150 pm, which is an estimate for the theoretical spectral resolution of the system. Moreover, the adoption of longer^[22] or specially designed fibers^[24–27] is expected to improve the resolution.

The spectrum of incident light simultaneously affects both the number of guided modes and propagation constants, while the polarization only influences propagation constants. Therefore, speckles under different polarization states show similarity, resulting in higher correlation coefficients. To demonstrate this, speckle patterns corresponding to various polarization states are obtained, including left-hand circular, right-hand circular, horizontal linear, vertical linear, 45° linear, and −45° linear polarized, all at a fixed wavelength of 1550 nm. The obtained correlation matrix of these six speckles is shown in Fig. 2(b). The lowest correlation coefficient is 0.46, which is higher than the correlation coefficient that can be achieved between different spectra, as shown in Fig. 2(a). The correlation coefficient may not comprehensively reveal the relationship between speckles under different states, and even with the same correlation coefficient, the influence of wavelength shift and changes in polarization states on the speckle pattern structure still differs. In practical experiments, the reconstructed results of speckles with different polarization states and the same wavelength hold the same spectrum, including a correlation coefficient of 0.46. The same situation applies to speckles with different wavelengths and the same polarization state. Therefore, for two speckle patterns with a correlation coefficient of 0.46, it still can be determined whether the correlation is caused by wavelength drift or changes in polarization state, and accurate spectrum reconstruction results can be acquired even though polarization has changed. Furthermore, precise polarization measurement can be achieved based on the differences in speckle patterns of various polarization states and the same spectrum.

The methods for obtaining the transmission matrices corresponding to the spectrum and polarization measurements are not consistent. The transmission matrix for the spectrum measurement is directly constructed from the speckle patterns of various wavelengths. Each column represents the intensity distribution of the speckle pattern at the respective wavelength. However, the transmission matrix for polarization measurement cannot be directly obtained from the speckle patterns corresponding to different polarization states. The essential distinction between polarization and spectrum lies in their superimposability: The spectrum can be sparse or dense, and the incident lightwave can be reconstructed using a matrix with fixed dimensions, while polarization states are mutually independent. Simultaneous measurement of a larger number of polarization states necessitates extending the dimensions of the transmission matrix. Therefore, the calibration process for polarization relies on establishing the correspondence between known polarization states and speckles, achieved through matrix inversion to derive the transmission matrix.

The impact of MMF parameters on spectral resolution has been thoroughly analyzed^[22], while their influence on polarization measurement performance requires further exploration. Previous studies^{[19,20,28–33]} mainly focus on the accuracy of the polarization measurement and overlook resolution. The polarization resolution signifies the capability to identify different polarizations. To explore this significant system index, we define polarization resolution as the minimum spherical distance between distinguishable polarization states on the Poincaré sphere, as shown in Fig. 3(a). The polarization resolution of the proposed system is 0.001437, indicating that for any polarization state on the Poincaré sphere, other states differing more than 0.001437 in spherical distance can be distinguished by the system. Adopting the spherical distance formula θ=Rarccos[cosφ1cosφ2cos(λ2−λ1)+sinφ1sinφ2], where S represents the spherical distance, R is the spherical distance, η denotes the ellipticity angle, and α is the orientation angle, the system is capable of identifying two polarization states with a polarization deviation angle greater than 0.51°.

To quantify the influence of the length of MMF on polarization resolution, simulations are performed by fixing the NA and CD of MMF at 0.22 and 105 µm, respectively. To illustrate the universality of this scheme, the fiber used in the practical experiment is one of the most common commercially available MMF, whose NA and CD are 0.22 and 105 µm, respectively. The 2 m length is chosen for good measurement performance while maintaining fine stability. Besides, to obtain credible data to guide the experiment and ensure consistency, the optical fiber parameters in the simulation are numerically selected to match those of the fiber used in the experiment. Changes in the length of MMF cause variations in the coupling between modes as described in Eq. (1), leading to alterations in the polarization resolution. As shown in Fig. 3(b), the polarization resolution of the proposed system increases with the MMF length. The increase in the MMF length results in a reduced correlation between speckles under different polarization states, thereby enabling the system to identify smaller differences between polarizations.

To further verify the impact of the NA of MMF on polarization resolution, the length and CD of MMF are fixed at 2 m and 105 µm for the simulation, respectively. Following the calculation, the respective polarization resolutions for various NA are shown in Fig. 3(c). The decrease in NA reduces the number of guided modes and changes in propagation constants, leading to increased correlation coefficients between speckles under various polarization states. Consequently, the polarization resolution worsens. NA cannot increase infinitely due to physical reality, and consequently, the polarization resolution can only increase finitely with NA.

Practical experiments often encounter limitations in fully exciting all modes supported by MMF due to factors such as the coupling method. To further test the impact of the number of the guided modes in MMF on polarization resolution, the length, NA, and CD of MMF are fixed at 2 m, 0.22, and 105 µm for simulation. The simulation result is shown in Fig. 3(d). The speckle correlation decreases with the increasing number of modes, and therefore the polarization resolution performance improves with the increase in the number of modes. The increase in the number of modes reduces the correlation coefficient between speckle patterns under various polarization states, thereby improving the polarization resolution performance.

In the experiments, we use a 2 m long step-index MMF (CD = 105 µm, NA = 0.22), and Fig. 4(a) shows the results of the simultaneous measurement for spectra and polarization information of incident lightwaves with an operational bandwidth of 30 nm. The upper and lower parts show the reconstructed spectra results and corresponding Stokes parameters, respectively. Figure 4(b) shows the spectral resolution. The proposed system successfully identifies two probe lightwaves (dashed line) with a separation of 100 pm. Figures 4(c)–4(e) show the distribution of measurement errors based on the reconstructed result in Fig. 4(a), including the angle of polarization (AOP), degree of circular polarization (DOCP), and degree of linear polarization (DOLP). The horizontal coordinate represents the measurement error, while the vertical one represents the corresponding percentage. The errors of reconstructed results for three Stokes parameters are 3.37%, 1.01%, and 0.84%, respectively, and the overall MSE is 5.3×10−5.

A trade-off relationship exists between resolution and stability. Using long MMF helps to improve measurement resolution, while long fibers are sensitive to external perturbations. Both vibration and temperature influence the transmission characteristics of the fiber, affecting the measurement performance. The fibers used in the system are fixed by tapes and placed in an open environment during the experiment. To characterize the stability of the system, the spectrum and polarization state are fixed, and the changes in the speckle correlation coefficient over 30 min are shown in Fig. 5(a). The correlation coefficients between the speckles at various times and the initial state remain over 0.98, indicating fine stability. The external temperature is the main factor that affects the stability of the system over a long time. Improvement in stability is expected to be achieved by using more advanced temperature control tools.

A calibration process needs to be performed to obtain a transmission matrix before practical measurement, and variations in the number of speckles sampled during calibration impact the polarization measurement performance, as shown in Fig. 5(b). The number of speckles sampled during calibration represents the number of polarization channels in the transmission matrix. As the number of polarization channels increases, the speckles of the incident lightwave become more correlated with the speckles stored in the transmission matrix, leading to decreased reconstruction error. However, excessive polarization channels cause the speckles of the incident lightwave to become highly correlated and difficult to identify with the stored speckles in the transmission matrix, and therefore the reconstruction error rises slightly. The error does not rise much because a redundant transmission matrix does not cause reconstruction failure.

Factors including ambient lightwave, wavelength drift, and intensity variations of the laser^[22] introduce experimental noise, and noise deteriorates the reconstruction accuracy by reducing the speckle contrast. To simulate the effect of unavoidable noise in the experiment, we add random noise to the speckles. The signal-to-noise ratio (SNR) is defined as SNR=10lg(Is/In), where Is and In represent the intensity of signal and noise, respectively. The MSE of polarization measurement as a function of SNR is shown in Fig. 5(c). The result shows that the system can achieve reconstruction with high accuracy at high noise level (SNR≥6dB).

The proposed scheme has the potential to measure both completely polarized and partially polarized lightwaves. However, limited by the equipment adopted in the system, the measurement for partially polarized lightwaves cannot be validated directly. Measurement of partially polarized light requires additional information about the intensity to calculate the degree of polarization (DOP=S12+S22+S322/S0), and partially polarized light can be described as the superposition of a completely unpolarized lightwave and a completely polarized one. The completely unpolarized lightwave does not provide polarization information and acts like noise, deteriorating the measurement accuracy by reducing the speckle contrast. Therefore, based on the Stokes parameters acquired by the system, DOP can be calculated from the measured lightwave intensity. Since the CCD responds differently to different light intensities, the intensity of the incident lightwave can be determined by weighting and integrating the intensity distribution of the speckle obtained by CCD. As shown in Fig. 5(d), there is a linear relationship between the intensity of the incident lightwave and the intensity calculated by weighted integration of the speckle, suggesting that the intensity of the lightwave can be measured by integrating the speckle.

The comparison of the proposed system with other spectropolarimeter schemes is shown in Table 1. Compared to other works, the proposed system demonstrates high spectral and polarization resolution with low polarization reconstruction error. Further improvement of measurement performance can be achieved by adopting more advanced reconstruction algorithms including neural networks, which may enable the potential measurement of additional lightwave field properties such as the object depth and multiple orbital angular momentum.

In conclusion, we propose a novel single-shot all-fiber computational spectropolarimeter based on a speckle pattern, enabling high-resolution and high-accuracy simultaneous measurement of the spectrum information and polarization state for incident near-infrared lightwaves. The proposed system achieves a spectral resolution of 100 pm with an MSE of 5.3×10−5 for polarization measurement. The errors of reconstructed results for the three Stokes parameters are 3.37%, 1.01%, and 0.84%, respectively. In addition, we propose a concept of the polarization resolution, evaluating the performance of polarization measurements by considering the precision and accuracy of the experimental results together. The polarization resolution of the proposed system reaches 0.001437. This high-performance spectropolarimeter shows a prospect of high-resolution multi-parameter measurements for circular dichroism spectroscopy, integrated transmission and sensing in fiber, and biomedical imaging.