Spectrometers play an indispensable role in modern industries and scientific research, including physics, biology, and medicine [1–5]. However, traditional high-resolution spectrometers suffer from several drawbacks, such as high cost, heavy weight, and large size, which limit their application in specialized environments. As a result, there is a growing demand for miniaturized, low-cost spectrometers with high resolving power. In recent years, advancements in disordered photonics [6,7] and computational algorithms [8,9] have enabled the development of speckle-based reconstructive spectrometers. These speckle spectrometers replace bulky dispersive elements with a compact scattering medium, enabling spectral detection by offering a mapping from incident wavelength to output intensity profile. Various scattering elements have been employed, including integrating spheres [10–12], frosted glasses [13,14], waveguides and fibers [15,16], nano-void chips [17], natural pearls [18], and so on.

Compared with other scattering media schemes, the fiber-based speckle spectrometer has advantages such as lightweight design, simple structure, ease of use, and straightforward multiplexing capabilities. In a fiber-based speckle spectrometer, the resolution is proportional to the temporal spread of light as it propagates through the optical fiber [19,20]. The temporal spread can be equivalently described as the distribution of optical path lengths, determined by the difference between the shortest and longest optical paths within the fiber. This difference is quantified as the product of the fiber’s geometrical length and the variation in propagation constants between the highest-order mode and the fundamental mode [19,20]. Therefore, an intuitive approach to enhancing resolution is to increase the fiber length. For instance, Cao et al. [21] achieved a spectral resolution of 1 pm using a 100-m-long multimode fiber (MMF) as the scatterer. However, system drift and environmental disturbances make such long fibers highly unstable. When the fiber length is reduced to 4 cm, the spectral resolution drops sharply to 2 nm. To balance high resolution and compactness, many studies have turned to exciting higher-order modes. Transverse mode control strategies include breaking the circular symmetry of the fiber through offset splicing [22] and introducing strong mode coupling effects via tapered structures [23,24], though these often come at the cost of ∼1% detected photon efficiency. Very recently, Guan et al. [25] adopted a periodic cascade of coreless fibers and photonic crystal structures, as well as a periodic tapered coreless fiber design [26], to disrupt mode transmission and effectively excite higher-order modes. These works explore the longitudinally controllable degrees of freedom in fibers, but such artificially periodic structures require multiple splicing or fabrication procedures.

Rather than engineering disorder, we focus on leveraging intrinsic disordered structures in a straightforward manner. In this study, we turn our attention to fused fibers. The catastrophic fuse effect was initially witnessed in silica single-mode fibers (SMFs) [27] and has since been demonstrated in a variety of other fiber types [28–30]. This effect is triggered when high-power optical signals are introduced into fibers experiencing mechanical or structural imperfections, such as damaged connectors, sub-optimal polishing, or excessive bending. Once initiated, the fuse effect becomes a self-sustaining process, characterized by the backward propagation of an optical discharge along the fiber core. This discharge generates intense localized heating, resulting in permanent structural damage to the fiber. A distinctive feature of this phenomenon is a trail of voids within the fiber core. Remarkably, this unique void pattern has opened new avenues for research, particularly in the development of innovative in-fiber sensor technologies [29,31,32].

In this paper, we recycle a destroyed fiber caused by the catastrophic fuse effect and utilize it for a low-cost speckle spectrometer. The damaged tracks in the fiber scatter the transmitted light into cladding modes and result in an intricate speckle at the distal end. This speckle pattern serves as a unique “fingerprint” for a given wavelength and is experimentally demonstrated to achieve a sensitivity of 0.1 nm. This performance metric, which previously required a 2-m-long conventional MMF, is now obtained with a 20-fold reduction in fiber length. We further propose a spectral reconstruction framework that involves a weighted transmission matrix, automatic differentiation (AD), and the gradient descent optimization. In contrast to traditional Tikhonov regularization, our method realizes a 48.93% reduction in error while boosting the peak signal-to-noise ratio (PSNR) by 6.12 dB.

In the fused-fiber speckle spectrometer, low-order mode light propagating through the core is scattered by the longitudinal distributed cavities, resulting in coupling into the fiber cladding. The interference among these scattered cladding modes produces a speckle pattern that varies with wavelength. For monochromatic input light, the electric field at the fiber’s output is the superposition of contributions from all guided modes: E(r,θ,λ,l)=∑mAmψm(r,θ,λ)exp[−i(βm(λ)l−ωt+φm)],(1)where Am and φm represent the amplitude and the initial phase of the mth mode, corresponding to a spatial profile ψm and propagation constant βm. l denotes the fiber length. Variations in the input wavelength λ modify the propagation constant β, causing the accumulation of phase delays βm(λ)⋅l and thus leading to a global translation of the speckle pattern.

In previous speckle-based reconstructive spectrometers, one approach to harness multimode interference patterns for spectral measurement is using MMFs [33,34]. In MMFs, there is no strong scattering induced by axial refractive index modulation; thus modes are confined to propagate within the core. To highlight the uniqueness of our approach, we employ the beam propagation method [35] to numerically simulate the field distribution along the length of different fiber structures. Figure 1(a) illustrates the three types of fiber configurations considered in this study: SMFs with and without micro-cavities (core diameter Dcore=26μm, cladding diameter Dcladding=400μm, NA=0.045), and MMFs without micro-cavities (Dcore=400μm, Dcladding=440μm, NA=0.045). The parameters of the SMF match those used in subsequent experiments, which are discussed later. In our simulations, the refractive indices of the core and cladding are set to 1.4507 and 1.45, respectively. Based on existing literature [32,36], the damage air voids are modeled with a transverse radius of 5 μm, a longitudinal length of 7.5 μm, and a refractive index of 1. The periodicity of the air voids varies as a function of the light intensity in the core during the damage process. Therefore, the air void periodicity in the simulations is set between 10 and 15 μm to account for this variation. All fibers are straight and have a length of 10 cm. All fibers are excited by a Gaussian beam with a mode field radius of 5 μm and a wavelength of 1550.00 nm, which is normally incident at the center of the fiber core.

Figure 1(b) illustrates the 2D intensity propagation diagram of the fused fiber along the xz-plane. Due to the presence of axially disordered cavities, light undergoes multiple scattering events, escaping from the core into the cladding and forming a random array of bright and dark spots at the output end. Figure 1(c) displays the corresponding output speckle pattern and its intensity probability density function. Figures 1(d) and 1(e) present the simulation results for a fiber without micro-cavities. In this case, most of the optical energy remains confined within the core, resulting in a Gaussian-like spot at the output. Figures 1(f) and 1(g) show the simulation results for an MMF. Even though the core size of the MMF matches the cladding size of the fused fiber, the speckle pattern generated by modal interference exhibits less complexity. We attribute this to three factors: first, due to the small-mode-field excitation source, only modes localized in the core’s central region are initially launched; second, the MMF lacks longitudinal refractive index modulation that can enhance structural disorder; third, the MMF is only 10 cm long and straight in shape, which also reduces the excitation of higher-order modes and the energy coupling between modes.

In this section, we present experimental investigations of the fused-fiber-based speckle spectrometer. Figure 2(a) illustrates the experimental setup. When a high-power laser beam is injected, fiber damage is initiated by localized overheating at the fusion splice point on the distal end (see more preparing details in Appendix A). A segment of fused fiber is then repurposed to serve as the scattering medium. It has a length of 10 cm, a core diameter of 26 μm, a cladding diameter of 400 μm, and an NA of 0.045. Its light transmittance is measured to be 13%. Light from a tunable laser (linearly polarized, tuning range: 1520–1567 nm, 3 dB linewidth: 5 MHz) passes through a polarization-maintaining SMF to ensure a consistent spatial profile and polarization of the input light into the fused fiber. Figure 2(c) displays an image of the catastrophic damage caused by the fuse effect, revealing distinct bullet-shaped voids within the fiber. The speckle pattern at the output end of the fused fiber is magnified by a 50× objective lens and projected onto the sensor of a near-infrared camera. The camera exposure time is set to be 50 ms. Figure 2(b) shows the experimentally captured speckle images at wavelengths of 1550.00, 1550.05, and 1550.10 nm. These patterns exhibit fine granular features, consistent with the simulation results presented earlier. Figure 2(d) depicts the one-dimensional intensity distribution along the central row of the speckle image, demonstrating quasi-random fluctuations. Notably, when the wavelength separation reaches 0.1 nm, the differences in the intensity profiles become distinguishable.

The spectral correlation function of the speckle characterizes its sensitivity to variations in wavelength and is typically expressed as [19] C(Δλ,x)=⟨⟨I(λ,x)I(λ+Δλ,x)⟩λ⟨I(λ,x)⟩λ⟨I(λ+Δλ,x)⟩λ−1⟩x.(2)

Here, I(λ,x) represents the intensity detected at position x for wavelength λ, and ⟨ ⟩ denotes taking the average over spectral or spatial dimensions. The spectral resolution can be approximated by determining the half-width at half-maximum (HWHM) of C. To ensure statistically meaningful results, we characterize the spectral correlation curves using three independently prepared fused fibers, as shown in Fig. 2(e). The maximum observed HWHM value (0.1 nm) is demonstrated as the spectral resolution of this scheme. This resolution is comparable to that achieved by traditional speckle spectrometers, which typically require approximately a 2 m MMF [38]. In contrast, we achieve this performance using only a 0.1 m fiber, reducing the system length by a factor of 20. Moreover, this resolution is about six times higher than that of the same length MMF (0.6 nm) [25]. It is important to note that the effective wavelength resolution in practical spectral recovery also depends on factors such as the reconstruction algorithm [11,39] and the number of sampling channels [40,41]. Different reconstruction methods can yield varying resolutions, more accurately referred to as “algorithmic resolution.” The resolution improvement highlighted in this study is defined solely by the HWHM of the spectral correlation function, a metric intrinsically tied to the system’s hardware design. This universal metric is thus more appropriately termed “optical resolution.”

In addition to spectral resolution, stability is a critical factor for speckle spectrometers. In previous works, achieving high resolution typically requires long fiber lengths, making the system susceptible to environmental disturbances. In contrast, the fused fiber used in our system has a length on the centimeter scale, significantly reducing such influences at the hardware level. At the post-processing level, pixel binning has also been demonstrated to enhance the stability of speckle patterns [17]. Figure 3(a) shows the speckle stability curve of the fused-fiber spectrometer. During the stability monitoring process, the experimental system is not placed under specialized sealed protection. A laser with a fixed wavelength is injected into the fused fiber, and the raw speckle data obtained is first down-sampled using a convolution kernel with a side length of 13 pixels to achieve binning. Each point on the curve is calculated by determining the correlation coefficient between the speckle at the current time and the initial speckle at time t0. The total monitoring duration is 20 h, with intervals of 5 min. Over time, the correlation coefficient between the current speckle and the initial speckle decreases but remains above a high level of 0.96.

Although the output speckle of the fused fiber undergoes degradation over time, Fig. 3(b) shows a high correlation between two adjacent speckle images captured by the camera along the time domain. Occasional sharp drops in correlation may originate from intensity jittering of the light source. For the majority of the monitoring period, the adjacent correlation remains above 0.9980. This observation is significant, as it reveals that the system changes gradually. We also investigate the correlation between the speckle pattern at an intermediate time point and the patterns before and after it, as shown in Fig. 3(c). Regardless of whether the time shift occurs before or after the reference moment, the correlation coefficient increases with the magnitude of the shift and exhibits a nearly symmetric distribution centered around the reference moment.

Building on the above analysis of speckle temporal evolution, we propose a spectral reconstruction method based on a weighted transmission matrix. The reconstruction for spectrum S can typically be recast as the following optimization problem: minS∥I−T·S∥2⏟F(S)+R(S),subjecttoS>0,(3)where I is the speckle intensity, T is the transmission matrix, and ∥∥ denotes taking the norm. The first term F(S) in the objective function represents the data fidelity term, while the second term R(S) is a regularization term. Notably, the choice of T in F(S) plays a critical role in the reconstruction quality. A transmission matrix T that can accurately map S to I ensures that the spectral estimate S^, corresponding to the minimum of F(S), is accurate. Otherwise, minimizing F(S) does not guarantee that the spectral estimate is consistent with the reference value. As shown in Fig. 3(d), the transmission matrix varies with spatial parameter r, wavelength parameter λ, and temporal parameter t. The dark blue border represents the pre-calibrated transmission matrices before the measurement time ti, while the red border indicates the post-calibrated transmission matrices after ti. At the moment ti, there exists a transmission matrix T(r,λ,ti) (denoted by the yellow dashed box) that cannot be accessed directly because measurement is being performed. In previous studies [21,26,42], the T(r,λ,ti) is often assumed to be identical to the pre-calibrated one. This assumption, however, neglects the degradation of the scattering medium over time, leading to significant deviations in the recovered results. In this study, we closely approximate T(r,λ,ti) by utilizing the pre- and post-calibrated transmission matrices synchronously, as shown in the following formula: T(r,λ,ti)=∑k≠iakT(r,λ,tk).(4)

Here, ak represents the weighting coefficients corresponding to different time points tk. Similar to commercial spectrometers that offer recalibration options, our method allows users to freely decide whether and how often to recalibrate. To provide a comprehensive reference for these decisions, we conduct a detailed investigation of the impact of transmission matrices at different time shifts and various weighting coefficient combinations on the accuracy of spectral restoration (see Appendix G). Relative L2-norm error (μ) and PSNR, two commonly employed metrics [41,43] in this field, are used here to evaluate the accuracy of the reconstructed spectra with respect to the reference spectra (see Appendix E). By comparison, we found that the reconstruction accuracy is highest when using the average of the two transmission matrices closest in time before and after the measurement as T in the F(S) term (see Appendix G).

After modifying the objective function in Eq. (3), we proceed to minimize it. In this study, spectral reconstruction is divided into two stages: rough estimation and efficient refinement [see Fig. 3(e)]. First, the regularizer R(S) is set to γ∥S∥2, corresponding to the Tikhonov regularization (details provided in the Appendix F). Here, γ is the regularization coefficient, selected using the generalized cross-validation method [2]. After applying GCV-regulated Tikhonov regularization, we obtain an initial spectral estimate S^ini. This estimate is typically close to the reference value and is widely adopted [2,44], but it can be further refined for improved accuracy. Several iterative algorithms, such as simulated annealing [19,44] and particle swarm optimization [25,26], have been used for fine-tuning the spectra. However, these methods rely on randomized global search strategies, which limit their optimization efficiency. In this work, after obtaining S^ini, we optimize it using AD and gradient descent optimization (details provided in Appendix F) until the objective function converges, yielding the final reconstructed spectrum, as shown in Fig. 3(e). AD is a high-efficiency technique for computing derivative information. It has been extensively applied in deep learning frameworks and has gradually gained traction in the field of optics recently [45,46]. In Appendix I, we compare the accuracy of spectral reconstruction with and without the use of AD and gradient descent. In summary, our method achieves a 48.93% decrease in reconstruction error and promotes the PSNR by 6.12 dB relative to pure Tikhonov regularization.

To demonstrate its capability for diverse spectral reconstruction, the proposed spectrometer is characterized using several typical spectrum types as a proof-of-concept. First, a dual-peak spectrum is employed to evaluate the spectrometer’s resolution. The experimental procedure for generating a dual-peak spectrum is based on the method in the literature [23] (see more details in Appendix J). Figure 4(a) demonstrates that the spectrometer can resolve spectral lines separated by 0.1 nm, which aligns well with the estimated resolution shown in Fig. 2(e). In the dual-peak experiment, as the two wavelengths approach each other, their spectral peaks overlap and become indistinguishable (according to the Rayleigh criterion). However, when measuring tunable single lines with a step size of 0.02 nm, the proposed spectrometer recovers their peak wavelength positions with high fidelity, as shown in Fig. 4(b). Next, we evaluate the broadband operational capability of the spectrometer within the range of 1525–1565 nm. Lasers with a full width at half maximum (FWHM) linewidth of 0.04 pm, tunable over a 40 nm range, are accurately retrieved, as depicted in Fig. 4(c). The average recovered FWHM linewidth is approximately 0.11 nm, with an average PSNR of 33.94 dB across the entire operational range [see Fig. 4(d)].

In addition to measuring discrete spectral lines, we also consider continuous broad spectra. Figure 4(e) presents the reconstruction of a dual-channel spectrum with equal amplitudes (FWHM linewidth=2.5nm for each channel), filtered through an acousto-optic tunable filter (AOTF) from an amplified spontaneous emission (ASE) source. The side lobes, which are ∼10dB weaker than the main peak, are clearly resolved, with a relative L2-norm error of 0.1142 and a PSNR of 29.44 dB. By configuring the AOTF to five filtering channels, a spectrum with an FWHM linewidth of 10 nm and non-uniform amplitude is generated. Its reconstruction results are shown in Fig. 4(f), with a relative L2-norm error of 0.1154 and a PSNR of 26.88 dB.

In this work, we have demonstrated a high-resolution, compact speckle spectrometer utilizing a fiber damaged by the catastrophic fuse effect. By repurposing the intrinsic micro-structure of the fused fiber, we achieved a spectral resolution of 0.1 nm with only a 10 cm fiber length, a significant enhancement in compactness compared to traditional MMF-based systems requiring 2 m. Moreover, this resolution demonstrates an approximately sixfold enhancement over that of a same length MMF. The axially distributed micro-voids along the fused fiber enhance the optical frequency sensitivity without the need of careful treatment. Additionally, by transforming abandoned fiber into a functional component, the cost of the core dispersive medium is substantially lower. Based on the analysis of the temporal evolution of the fused fiber, we furthermore propose the weighted transmission matrix to modify the objective function for optimization. Inspired by deep learning frameworks, we employ AD and gradient descent for further spectral refinement. Our approach dramatically reduces reconstruction error by 48.93% and increases the PSNR by 6.12 dB compared to pure Tikhonov regularization. The spectrometer reconstructs various types of spectra across the 1525–1565 nm range, including dual-peak, tunable single-line, and continuous broad spectra, showcasing its versatility and accuracy. In this work, the 40 nm measurement range reported is limited by the maximum tunable range of the laser used for pre-calibration. The fused fiber spectrometer theoretically supports a broad wavelength range from 400 to 2400 nm, leveraging silica fiber’s wide transparency window [23].

Last but not least, it is necessary to clarify the respective roles of hardware and software in this study, as well as the interplay between the two. First, the fused fiber enables high spectral resolution, defined as the minimum distinguishable wavelength separation. Simultaneously, this fiber resolves the trade-off between resolution and size. Its compact length makes it less susceptible to external disturbances. Our post-processing strategy, which incorporates a weighted transmission matrix, is not proposed in isolation but tightly dependent on the hardware’s robust characteristic. The high fidelity of the weighted transmission matrix further provides the confidence to embed it into the loss function of the AD and gradient descent, ultimately yielding results with high accuracy. Unlike resolution, accuracy refers to the deviation between the reconstructed spectra and the reference. In summary, this study represents a novel integration of micro disordered media and advanced computational algorithms, providing a promising solution for miniaturized, lightweight, cost-effective, and high-performance spectral measurement technologies.

Acknowledgment. Junrui Liang thanks Cheng Yang and Hanshuo Wu for their help in this work.

Author Contributions.P.Z. and J.Y.L supervised the project. P.Z., J.M.X., and J.R.L. proposed the idea and designed the experiments. J.R.L. prepared the fused fiber. J.R.L. and Z.M.H. carried out the experiments. J.R.L. and J.L. proposed the reconstruction method. J.R.L. and J.H.H. conducted the simulation and experimental analysis and wrote the manuscript. P.Z., J.M.X, J.L., and J.Y. contributed to manuscript reviewing. All authors participated in the discussion of results.