Spectrometers are pivotal in diverse industries and scientific fields, including biology, astronomy, medical treatment, and laser beam characterization^[1–5]. Traditional bench-top spectrometers often face a dilemma between resolution and size. Over the past decade, there has been an increasing fascination with speckle-based reconstructive spectrometers (RSs) due to their novel computational mechanism and outstanding performance^[6–10]. Speckle-based RSs have been achieved using scattering mediums to encode spectral signals into spatial intensity patterns^[11–15]. Using interference among guided modes in a multimode fiber (MMF) is an alluring alternative^[16–19], considering the small footprint associated with its all-fiber structure and high resolution.

Speckle-based RSs bring advantages such as improved resolution and reduced size while facing urgent demand to improve accuracy^[19]. The accuracy is one of the key performance indicators of spectrometers. For traditional spectrometers, it depends on the quality of dispersion elements, the performance of detectors, the alignment accuracy of the optical path, etc., which have been well developed. However, for speckle-based RSs, improving the accuracy is still very challenging. Among existing schemes, the spectra are encoded to a speckle pattern by the transmission matrix (TM) of the MMF and then decoded by the reconstructive algorithm. For observed speckle intensity ${\mathbf{I}}_{N\times 1}$, TM ${\mathbf{T}}_{N\times M}$, and spectrum ${\mathbf{S}}_{M\times 1}$, the forward encoding can be represented as^[16–19]${\mathbf{I}}_{N\times 1}={\mathbf{T}}_{N\times M}·{\mathbf{S}}_{M\times 1}$, where $N$ and $M$ represent the spatial and spectral channel numbers. However, spectral encoding usually suffers from information loss and noises in transmission and imaging, which restrict the further improvement of reconstruction accuracy.

To minimize information loss caused by encoding, increasing the independent spatial channels of the observed $\mathbf{I}$ has earned sustained attention. Detecting high-dimensional optical field information from output speckle patterns is a viable approach, involving polarization^[20,21] and phase detection^[22,23] in addition to intensity-only measurements. However, detecting high-dimensional optical fields usually relies on bulky polarization beam splitters (PBSs) or complex interferometric setups, preventing the all-fiber structure of the system. Moreover, the freedom to expand informative observations is limited; for instance, the random distribution of output speckle full Stokes parameters already effectively characterizes its polarization features^[21]. To enhance the compactness of the RSs and further expand the meaningful information, manipulating and multiplexing the input end of the medium represents a promising method. Wavelength division multiplexing (WDM)^[24], space division multiplexing (SDM)^[25], and mode division multiplexing (MDM)^[26] techniques have been adopted to switch launching conditions at the input end of the mediums. Nevertheless, the expansion of WDM relies on high-cost devices with multiple channels, while the maturity of the components relied on by SDM and MDM is relatively low. In contrast, the polarization degrees of freedom at the input port have not been exploited. Polarization division multiplexing (PDM) technology^[27,28] enables the simultaneous transmission of two independent data streams with two mutually orthogonal polarization states, aiming to develop the overall system capacity. In an all-fiber speckle-based RS, by adjusting the initial input condition^[29,30] to two such polarization states, the transmission process of two beams in the MMF will be completely different, thereby easily increasing the amount of independent observed information.

Unlike traditional spectrometers that directly map each discrete spectral intensity to a pre-specified sensor unit, RSs encode all spectral information to an intensity image. The observed speckle pattern always contains various noises, leading to an accuracy reduction in spectral recovery^[31]. Inspired by bionics^[32], the idea of using the differential of two orthogonal polarization images has attracted the attention of researchers, which has shown significant effectiveness in background removal and enhancement of target visibility in scatter imaging^[33,34].

In this Letter, we present an all-fiber high-accuracy RS based on PDM technology with differential speckles called differential polarization division multiplexing (DPDM). To create extra light propagation routes, two orthogonal linearly polarized light beams are launched into the MMF, more comprehensively encoding the optical spectra into two speckles. Then, a differential processing of these two patterns is introduced to suppress the noise and improve the contrast. By decoding from the differential speckles, spectral measurement with 2 pm resolution and 2000 spectral channels can be achieved. In contrast to traditional TM and PDM schemes, the combination of the differential processing and PDM approach has a significantly lower spectral reconstruction error. Our scheme is simple, effective, and well-suited for versatile speckle-based RSs needing high accuracy.

The experimental setup of the proposed all-fiber high-accuracy RS is depicted in Fig. 1. A tunable laser (tuning range: 1520–1567 nm, 3 dB linewidth: 0.04 pm) in conjunction with a piece of single-mode fiber (SMF) was used for precise wavelength controlling. The SMF was fused with a single-mode optical fiber PBS. In this proof-of-concept experiment, we simply used PBS, which provides dual linear polarization. Moreover, this degree of freedom of manipulation on polarization can be rapidly extended using off-the-shelf motorized polarization controllers. After passing through the PBS, the light was divided into two beams of orthogonal linearly polarized light, then connected with a piece of step-index MMF (length = 100 m, core diameter = 105 µm, numerical aperture = 0.22). The MMF was tightly coiled and placed in a sealed box to counteract environment disturbances (see the Supplementary Information). Given the extended optical path, the interference between the guided modes within the MMF led to wavelength-dependent speckle patterns. These near-field speckle patterns were magnified using a $50\times$ objective lens (OL) and recorded by an infrared camera with a pixel size of $20\mathrm{\mu m}\times 20\mathrm{\mu m}$. Moreover, an attenuator (ATT) was adopted to prevent camera saturation before image acquisition.

During the calibration stage, the tunable laser wavelength was systematically scanned in increments of 1 pm across a range of 2 nm. This scanning process generated a spectral sequence comprising 2000 channels. Subsequently, during the measurement stage, the speckle pattern of the objective spectra was created by incoherently superimposing calibrated speckle patterns at different wavelengths^[18]. Generally, once the synthesized speckle pattern and the TM were provided, the spectra were reconstructed using the truncated inversion technique and the simulated annealing algorithm mentioned in Ref. [20]. The truncation threshold for matrix inversion was set to be ${10}^{-4}$, and the loss function of the optimization was ${\Vert I-I·S\Vert }_2^2+\gamma {\Vert S\Vert }_2^2$, where ${\Vert \Vert }_2$ represents the L2-norm and $\gamma$ represents the regularization coefficient with a value of 0.05.

To obtain differential speckles, we calculated the difference between the two speckles at the output end of the MMFs and used the absolute value, as illustrated in Eq. (1): $${\mathrm{Speckle}}_{\mathrm{\Delta }}=|{\mathrm{Speckle}}_{\perp \mathrm{incidence}}-{\mathrm{Speckle}}_{\parallel \mathrm{incidence}}|.$$(1)

Figures 2(a)–2(c) show representative output speckles corresponding to the horizontally and vertically polarized incidences, and the differential speckle. The white annotations in the lower right corner of the speckles indicate the speckle contrast, defined as ${\sigma }_{\mathbf{I}}/⟨\mathbf{I}⟩$, where ${\sigma }_{\mathbf{I}}$ denotes the standard deviation of the speckle and $⟨⟩$ denotes the average across position ranges. The contrast values of Figs. 2(a) and 2(b) are 0.53 and 0.45, respectively. After the differential operation, the contrast can be increased to 0.90 due to noise filtering. The contrast values of single-polarization incident speckle and DPDM speckle for 2000 spectral channels in the 1550–1552 nm range were calculated. The statistical results showed that the average contrast of the MMF output speckle with horizontally polarized incidence was 0.52. For vertically polarized incidence, the average value was 0.44, while the average contrast of DPDM speckle was 0.92. When measuring a spectrum, particularly when dealing with broadband signals, the resulting speckle pattern can be attributed to the cumulative effect of dependent speckle intensities from various channels. According to the demonstration in Ref. [20], when the individual wavelength channel exhibits a high degree of speckle contrast, the overall contrast of their accumulation will suffice for counteracting noise interference.

The PDM technology and differential speckles were employed for spectral reconstruction. Spectral reconstruction errors were calculated and compared in three cases: only one input launch condition (vertically polarized), both input launch conditions (vertically polarized and horizontally polarized), and differential operation of two input launch conditions. For the first case, it could be regarded as a conventional TM-based scheme. For the last two, we simply represented them as a PDM scheme and a differential PDM (DPDM) scheme.

In this paper, the accuracy is featured by the reconstructed error $\mu$, which is defined as $$\mu =\sqrt{\frac{1}{M}{\sum }_{i=1}^M{(S_i^g-S_i^r)}^2}/{\sum }_{i=1}^M{(S_i^g)}^2.$$(2)

We first assessed the spectral resolution by detecting two wavelengths separated by 2 pm. Figure 3(a) shows that TM accurately determined the positions of two narrow peaks, but the reconstructed intensities were inaccurate, and the signal-to-noise ratio (SNR) was $\sim 7\mathrm{dB}$. The SNR is defined as $$\mathrm{SNR}=10\log (S_{\max }^r/S_{\max }^n),$$(3)where $S_{\max }^r$ denotes the maximum value of the recovered spectrum and $S_{\max }^n$ denotes the maximum value of the recovered noise (absolute error between the recovered signal and the ground truth). The spectral reconstruction error of the TM method was 0.0108. The PDM approach also accurately reconstructed the narrow peaks with a spectral reconstruction error of 0.0036, and the SNR was limited to $\sim 16\mathrm{dB}$. The DPDM approach achieved a reconstruction error of 0.0009, 91%, and 75% lower than the TM and PDM schemes, respectively. The SNR of the reconstructed spectrum was $\sim 23\mathrm{dB}$, significantly higher than the TM and PDM schemes. In addition, we have also confirmed that when the spectral correlation bandwidth in the scattering medium is relatively small ($\sim 0.1\mathrm{nm}$), DPDM can largely improve the resolution (given in the Supplementary Information).

Furthermore, a Lorentzian spectrum with an FWHM of 0.5 nm was set as the target to evaluate the broadband signal reconstruction performance. In Fig. 3(b), the spectra recovered by TM, PDM, and DPDM approaches are shown alongside the ground truth. Both reconstructed spectra of TM and PDM suffered from heavy noise, while the DPDM showed the highest agreement with the ground truth. The reconstruction error of DPDM was 0.0287, which was 85% and 73% lower than TM and PDM (0.1977 and 0.1050), respectively. Reconstruction errors of broadband light centered at 1550.25, 1550.5, 1550.75, 1551, 1551.25, 1551.5, and 1551.75 nm were calculated repeatedly 30 times, and the average values were taken. When all other conditions were held constant, and only the center wavelength of the test light was varied, the relative standard deviations of the reconstruction for the three methods were less than 2%. We believe such fluctuations can be safely disregarded. This was also reasonable, as each wavelength channel shared the same experimental configuration for calibrating and reconstructing. No specific wavelength channel was supposed to exhibit any peculiarities.

Figure 4 illustrates the spectral reconstruction errors obtained using different methods for spectra with different Lorentzian FWHM values. All methods had an increasing trend in reconstruction error as the FWHM increased. This could be attributed to the expansion of the detection bandwidth, which caused the intensity distribution of the corresponding speckle to become smoother, resulting in the disappearance of local features. The reconstruction became distorted when a significant amount of features vanished. It is worth noting that the DPDM scheme consistently had the lowest reconstruction error across all scenarios. Exactly, compared with TM (vertical polarization), TM (horizontal polarization), and PDM (vertical polarization and horizontal polarization), the average reconstruction error of the DPDM method decreased by 77%, 76%, and 69%, respectively. It also offered the advantage of faster calibration and reconstruction. Still considering the inverse problem solution for $\mathbf{S}$ with known variables of $\mathbf{I}$ and $\mathbf{T}$, $\mathbf{I}$ is $2N\times 1$, $\mathbf{T}$ is $2N\times M$, and $\mathbf{S}$ is $M\times 1$ in the PDM method, while $\mathbf{I}$ is $N\times 1$, $\mathbf{T}$ is $N\times M$, and $\mathbf{S}$ is $M\times 1$ in the DPDM method. In the calibration stage, taking spectral reconstruction with an FWHM of 0.5 nm as an example, the PDM method on average required 61 s to finish the TM inverse calculation. In contrast, the DPDM method only took 12 s on average, merely 20% of the time required by the PDM method. This would greatly benefit situations in which online TM calibration is needed. During the reconstruction stage, the PDM method required convergence in 178 s on average, whereas the DPDM method achieved a 66% cost reduction in only 60 s on average. This is of great interest as faster speed and higher accuracy can be realized solely by implementing simple differential operations. The calculations were repeated 30 times on a computer with a 2.1 GHz Intel Core i7 CPU and 16.0 GB RAM, without using GPUs.

The DPDM method was utilized to reconstruct various spectra. Figure 5(a) illustrates a series of reconstructed narrow spectral lines (FWHM = 0.1 nm) from 1550 to 1552 nm. The average spectral reconstruction error was 0.0131, and the SNR surpassed 10 dB in the working range. Figure 5(b) shows the reconstructed result for a continuous broadband random spectrum composed of overlapped Gaussian kernel functions. The spectral reconstruction error was 0.0310.

In summary, we developed an all-fiber high-accuracy RS using differential speckles based on PDM technology. To the best of our knowledge, it is the first time that PDM was adopted in RSs. It was realized by employing a PBS to launch two orthogonal linearly polarized light beams into the MMF. As a result, the transmissible information was enhanced by expanding the light scattering process; thus, the spectrum can be encoded more completely into a speckle pattern. A differential operation was proposed to resist measured noise and improve the contrast. The proposed RS showed that a spectral resolution of 2 pm with 2000 spectral channels could be achieved. Compared with traditional TM and PDM schemes, the proposed DPDM approach reduced the spectral reconstruction error by 77% and 69%, which is quite simple and effective for spectral measurement requiring high accuracy.