Deciphering the complexities inherent in scattering media holds paramount importance across a wide range of applications, spanning from the microscopic scale in biophotonics to the macroscopic realm in remote sensing. Traditionally perceived as random and intractable, optical scattering has long posed a formidable challenge to the effectiveness of numerous optical technologies, particularly in deeper exploration and imaging. However, a groundbreaking development has recently revolutionized our understanding: wavefront shaping.1^–3 This approach treats the scattering process as predictable rather than stochastic. By constructing a linear scattering matrix that connects the input and output planes, it becomes feasible to counteract the disorder induced by scattering. Through careful manipulation of light across spatial, temporal, and spectral dimensions, this methodology facilitates effective optical imaging, sensing, and manipulation even in highly scattering environments.4^,5 This innovation marks a significant leap forward, offering novel insights into the management of optical scattering and augmenting the capabilities of optical technologies in complex scenarios.6^–8

In addition to compensating for scattering effects, there has been a growing interest in understanding the continuous evolution of scattering characteristics. These phenomena, known collectively as the memory effect, are notable for their resilience to external perturbations and their critical roles in wavefront shaping applications.7^,9^–11 For example, the temporal memory effect associated with the scattering matrix establishes a timeframe during which dynamic changes occur, serving as a critical tool for effective wavefront modulation and dynamic tracking.12^–14 Likewise, spatial evolution, including angular and translational memory effects,15^–18 not only delineates the minimum sampling range but also lays the groundwork for novel computational methods in non-invasive imaging through scattering media.19^–22 Moreover, the rotational memory effect observed in multimode fibers contributes to imaging techniques by enhancing point-scanning speeds by mitigating frequent matrix recalibrations.23^–26 In all these applications, there exists a pervasive aspiration to augment the scope of these memory effects in both the temporal and spatial domains.27^,28

Compared with its temporal and spatial counterparts, the spectral domain, due to its extension of a predominant dimension for nonlinear manipulation, promotes their applications involving broadband illumination.29^–31 Analogous to scattering characteristics, the spectral memory effect is crucial in manipulating scattering over a broadband spectrum. Early investigations into spectral intensity fluctuations under passive illumination conducted in-depth analyses, culminating in a passive regime established through the Langevin approach.32^–34 However, passive analysis, without considering structured input wavefront excitation, cannot guide modern applications with active wavefront modulation. Initial efforts to actively manipulate broadband scattered light often relied on a brute-force approach, such as iteratively fine-tuning phase values based on feedback.35 The advent of the multispectral transmission matrix marked a significant stride forward. This concept recognizes that various frequency components are scattered in disparate manners, thereby facilitating the independent manipulation of these components and enabling the generation of ultrashort pulses tailored with specific spatiotemporal waveforms.36^,37 The fidelity of the reconstruction is heavily contingent upon the precise modeling of the spectral memory effect.38^,39 In previous studies, typically regarded as a passive consequence of the average influence of the scattering medium, the spectral memory effect’s range exhibits an inverse relationship with the time photons spend traversing the scattering medium itself,40 undergoing a sharp decline with inverse proportionality to the square of the depth.38^,39 Regrettably, those passive consequences emphasize a fatal blank of active manipulation. It means that in applications that require a specific spectral range or resolution with flexibility, the ability to actively manipulate this effect is highly desired. For instance, actively suppressing the spectral memory effect enables the precise discernibility of adjacent independent spectral responses without the need to passively increase scattering complexity. This capability is transformative for applications such as high-resolution spectral scanning and single-pixel spectrometers based on multimode fiber,41 where it can significantly improve resolution and accuracy in spectral analysis. By leveraging the spatial dimension compatibility of wavefront shaping, the active manipulation of the spectral memory effect provides a powerful tool for controlling spectral sensitivity, opening new possibilities for advanced optical systems. Conversely, enhancing the spectral memory effect can effectively mitigate the deterioration of spatiotemporal waveforms caused by intrinsic scattering in broadband illumination. This is particularly impactful in deep-tissue imaging and manipulation, where spatiotemporal interactions require precise nonlinear control. Applications such as multiphoton imaging and optical manipulation42^,43 stand to benefit greatly, as the method addresses the challenges posed by scattering in biological tissues. The simultaneous compensation and manipulation of spatial and spectral characteristics offer a unified approach to spatiotemporal wavefront shaping, paving the way for breakthroughs in biophotonics and nonlinear optical interactions.

Prior research has demonstrated that the deliberate redirection of light into specific eigenchannels44 can significantly influence physical parameters, including transmission, reflection, and absorption rates.45^–47 This approach can also customize the range of the angular memory effect through active manipulation, guided by theoretical analysis using random matrix geometry.48^,49 Given this, a pertinent question arises: can coupling light into different scattering eigenchannels effectively manipulate other physical effects? To address this question, the Eisenbud-Wigner-Smith operator has been developed, demonstrating effectiveness in manipulating the temporal and spectral parameters of scattered light.50 However, in practical scenarios, achieving comprehensive transmission derivation across the entire spectrum to construct such an operator presents significant challenges due to time constraints and the inability to perform calibration across the spectral domain. Consequently, there is a pressing need for a more efficient theoretical framework that can effectively manipulate the spectral memory effect, particularly in single-wavelength acquisition. To explore this, we have developed a theoretical framework that combines wave-based interference and photon-based propagation mechanisms. This framework implicitly parallels the early passive diffusion regime, conceptualized through a Feynman-like diagram involving the joint effects of scattering trajectory analysis and joint Green’s functions, which together elucidate the mechanisms underlying the spectral memory effect. Our approach reveals a robust positive correlation between the spectral memory effect and the transmission efficiency of scattering eigenchannels, challenging the long-held understanding that the spectral memory effect is merely a passive characteristic of the scattering medium. Through finite-difference time-domain (FDTD) simulations and experiments, we demonstrate that the range of the spectral memory effect can be actively manipulated, with variations in enhancement and suppression exceeding threefold (∼3.27).

Addressing the issue depicted in Fig. 1, we explore a scattering medium under broadband light illumination. Unlike the discussion of spatial evolution when coupling different spectra through transmission geometry at different chromato-axial planes,51^,52 considering the spectral correlation of scattering characteristics itself provides a deep insight into the spectral memory effect. The wavelength-dependent nature of scattering causes the input-output relationship in matrix representation, denoted as T(λ), to change significantly across wavelengths λ. This variance manifests in overlapping field distributions E(λ), of which the spatial variation can be adjusted spectrally. As appropriate facilitation to the theoretical modeling and subsequent rigorous verification, the intensity correlation I(λ)=|E(λ)|2 between speckle patterns at adjacent wavelengths λ and λ+Δλ with a definition of Cλ(Δλ)=⟨I(λ)I(λ+Δλ)⟩/⟨I(λ)⟩⟨I(λ+Δλ)⟩ will be adopted rather than E(λ) with hidden phase, where the angular brackets ⟨·⟩ signify an inner product over two-dimensional space. By conventional standards,39^,53 the spectral memory effect is quantifiably identified as the spectral correlation range Δλ away from the wavelength λ that reduces Cλ(Δλ) to 1/e. Our initial approach employs a diffusion model to examine the evolution of the scattering matrix. According to diffusion theory, light traverses the scattering medium via numerous paths, guided by a distribution L that is analytically determined by the medium’s physical attributes, such as diffusion constant and thickness. For the p’th path with distance Lp, the accumulated optical effect is expressed as exp[(−α+ik)Lp], where α represents the attenuation coefficient and k is the wavenumber at wavelength λ. Assuming a minor detuning Δλ (or Δk) does not significantly alter the distribution of paths or cause dispersion, the slight optical field adjustment is approximated by exp(iΔkLp)≈1+iΔkLp. The scattering matrix’s elements are derived from summing up all optical path contributions, leading to the relationship T(λ+Δλ)∝T(λ)[I+X(Δλ)].(1)

Here, I is the identity matrix and X is the detuning matrix induced by wavelength detuning. Their complex elements χ in X conform to a circular-Gaussian distribution χ∼N(0,σ2/N), where the norm of X is proportional to σ2/N and N is the matrix’s dimensionality. Because the value of σ embodies a close function to the detuning wavelength, we also define this standard deviation as a spectral detuning parameter σ. The formula σ=L¯Δk=L¯2πΔλ/λ2(2)indicates that X diminishes as Δλ nears zero. Moreover, the average path length L¯, determined by the medium’s scattering characteristics, also influences the spectral memory effect’s range. This observation aligns with prior research, suggesting a direct link between related properties in scattering medium and the control of the extent of the spectral memory effect.38^,39

Next, we explore the spectral memory effect across various scattering channels. By applying random matrix theory,45^,54 we decompose the scattering matrix using singular value decomposition: T=UτV†, where U and V are unitary matrices and τ is a diagonal matrix representing energy transmittance. Engineering an input wavefront as Vn (the n’th column of V) targets the n’th eigenchannel of the scattering medium, yielding an output wavefront of τnUn (the n’th column of U), with τn denoting energy transmittance. Through detailed mathematical analysis provided in Appendix A, we derive the intensity correlation coefficient for the activated eigenchannel as a function of detuning wavelength Cλ(Δλ)=Cn=τn(1+σ2/N)τn+τ¯σ2,(3)where Cn gives the spectral correlation modulated by the n’th eigenchannel and τ¯ represents the ensembled-averaged transmission across randomly selected scattering channels. For large N is large and small σ, this equation simplifies to 1−τ¯σ2/τn. This formula reveals that higher transmission eigenchannels with large τn exhibit slower correlation decay, whereas lower transmission channels with small τn show faster decay. These findings discover a direct correlation between the transmission of scattering eigenchannels and the range of the spectral memory effect, indicating that selective coupling of light into different scattering eigenchannels can actively manipulate this effect. In fact, these physical phenomena also suggest a similar clarification in ballistic trajectory activated by specified eigenchannel, supporting the resilience of wavelength change for short path lengths.

To validate our theoretical model, we employ FDTD simulations, focusing on a scattering medium comprised of randomly distributed high refractive index cylinders within a low refractive index homogeneous medium (scattering coefficient μs=2.703mm−1 without absorption). This simulation setup is consistent with our prior studies,55 with detailed configuration elaborated on in Appendix B. Upon measuring the scattering matrix and identifying eigenchannels (N=25), we analyze three representative correlation coefficients as a function of the detuning wavelength for high-, medium-, and low-transmission channels, as depicted in Fig. 2(a). It is noted that the quantification and the sequencing of the transmittance for coupling different eigenchannels are both carried out by the actual observation within the targeted area instead of the mathematical prediction. The results show a notably slower decay for the high-transmission channel. Moreover, we create a scatter plot for three representative detuning wavelengths Δλ=1, 3, 5 nm, selecting every other eigenchannel by increasing eigenvalues, resulting in a total of 13 scattering eigenchannels, illustrated in Fig. 2(b). For clarity, the highest transmission eigenvalue in each case is normalized to 1. This figure demonstrates that larger detuning wavelengths correlate with lower correlation coefficients, and there is a positive relationship between correlation coefficients and transmission eigenvalues. To confirm the precision of our defined spectral detuning parameter σ from Eq. (2), we first determine σ values by fitting to Eq. (3). These values are then plotted against the detuning wavelengths Δλ, as shown in Fig. 2(c). A linear fitting procedure confirms the close alignment between simulation results and theoretical predictions, thereby validating our model’s accuracy. Finally, Fig. 2(d) presents the determined range of the spectral memory effect for the 13 analyzed scattering eigenchannels. The results indicate that coupling light into eigenchannels with high transmission can significantly extend the range of the spectral memory effect. Due to the practical definition of τ¯, we reduce the quantitative fitting in Fig. 2(d). Nevertheless, the trend of function with increasing τn still shows in great accordance with the simplified representation Δλ∝τn. Quantitatively, with 40 independent controls, the ratio between the largest and smallest ranges is ∼6.95 to 3.18 nm, which is about 2.19. This demonstrates the substantial impact of selective light coupling into high-transmission eigenchannels on enhancing the spectral memory effect.

We further showcase our experimental setup designed to explore the spectral memory effect in Fig. 3, building upon our previously developed methodology for nonholographic measurements of the scattering matrix via phase retrieval.56^,57 During the phase retrieval, the native exclusion of the relative phase between individual speckles would not affect the characterization of eigenchannels. The setup employs a continuous-wave laser, offering a wavelength tuning range from 638 to 640 nm with a fine-tuning step size of 0.1 nm. This source enables precise control over the wavelength for detailed analysis of the spectral memory effect. Wavefront modulation is achieved using a reflection-mode spatial light modulator, which allows for phase-only modulation, crucial for accurately directing the wavefront through the scattering medium. The chosen scattering medium for these experiments is a 1-mm-thick resin block, embedded with ∼100-μm-diameter alumina powder, as detailed in Appendix C. For experiments conducted at the red-light wavelength of 639 nm, we measured the scattering and absorption coefficients as μs=0.74mm−1 and μa=0.04mm−1, respectively.

We measure the scattering matrix and characterize the eigenchannels (N=400) to investigate the spectral memory effect further, with results presented in Fig. 4. The number of eigenchannels N=400 is determined as an appropriate consideration of experimental limitations and technical performance for large-scale wavefront shaping.57 In Fig. 4(a), we display the correlation coefficients as a function of the detuning wavelength for three representative eigenchannels, where the detuning wavelength starts at 638 nm within correlation analysis. The fitting curves demonstrate that channels with high transmission exhibit a slower decay. In addition, for three selected detuning wavelengths Δλ=0.3, 0.8, 1.3 nm, we plot correlation coefficients for 41 eigenchannels across various eigenvalues in Fig. 4(b), revealing a trend consistent with FDTD simulation outcomes. A notable observation is the concentration of experimentally measured eigenvalues within a narrow range. A linear fit of the spectral detuning coefficient as a function of detuning wavelength, shown in Fig. 4(c), indicates a nonzero intercept. This nonzero intercept primarily stems from the inaccuracy of the scattering eigenchannels, a divergence attributed to the limitations of the experimental setup’s imperfections. Specifically, low-transmittance channels experience instability, especially in the thermal noise environment. Thus, the experimentally derived slope with accumulative intercept leads to approximately three times smaller than that predicted theoretically using Eq. (2) (slope ∼2.660). On the other hand, discrepancies due to model inaccuracies are also attributed to unavoidable absorption and finite bandwidth effects, which will be discussed subsequently. As the quantification of the spectral memory effect, Fig. 4(d) illustrates the range of spectral correlation across the 41 analyzed scattering eigenchannels. Using 400 independent controls, the ratio between the largest and smallest ranges of spectral correlation is ∼4.09 to 1.25 nm, equating to a ratio of about 3.27. These experimental findings reinforce the validity of our proposed model, highlighting its effectiveness in actively manipulating the spectral memory effect within complex scattering media.

To gain a deeper understanding of the effects of absorption in the scattering medium and the finite bandwidth of the illumination on the spectral detuning parameter σ with respect to detuning wavelength Δλ. Turning to FDTD simulations, these simulations allow for the isolation and individual examination of these factors, which are difficult to control in experimental settings. Absorption is modeled by increasing the imaginary part of the refractive index of the cylinders within the simulation, achieving an effective absorption coefficient μa=0.019mm−1. To simulate the finite bandwidth effect, we use illumination with a 2-nm bandwidth, diverging from the single-frequency light typically employed. It is important to note that the physical dimensions used in these simulations are significantly smaller than those in actual experimental conditions, a necessary compromise to reduce computational demands. Consequently, these physical parameters require rescaling for accurate correlation with experimental observations. As depicted in Fig. 5, the introduction of absorption and finite bandwidth notably decreases the slopes of σ as a function of Δλ. This effect indirectly extends the range of the spectral memory effect. The findings corroborate expectations,39^,58 as both absorption and finite bandwidth limit the penetration of long-distance traveling light within the medium. This mechanism qualitatively accounts for the observed discrepancies in the slope values between experimental outcomes and theoretical predictions, as seen in Fig. 4(c).

In summary, our study demonstrates that the spectral memory effect in scattering media can be actively manipulated by selectively coupling light into different eigenchannels. It is noteworthy that a previous study proposed a theoretical guess of resonance model to explain spectral properties of scattering eigenchannels in random media, predicting that the lower eigenchannels would correlate a wider spectral range in the pulse model with activated modes further away from resonance.59 However, by integrating concepts from scattering matrices and diffusion models, and merging insights from wave-based interference and photon-based propagation mechanisms, it seems that our theoretical framework gives an updated conclusion. In aspects of theory, simulations, and experiments, our model establishes a clear positive relationship between the spectral memory effect and the transmission efficiency of scattering eigenchannels. To strictly conform to the conditions of the theoretical structure, the moderate scattering thickness provides an ideal photon diffusion model, which would lead to a finite range of the spectral memory effect. Nevertheless, FDTD simulations and experimental data robustly support our hypothesis, illustrating significant alterations in the spectral memory effect’s range, with variations in enhancement and suppression exceeding threefold (∼3.27). In addition, we discover that factors such as absorption and finite bandwidth further enhance the spectral memory effect by limiting the propagation of long-distance traveling light within the scattering medium. This research lays a crucial foundation for advancing our understanding of scattering media under broadband illumination and provides a valuable tool for actively manipulating the spectral memory effect. This capability offers substantial benefits to various fields, including imaging, sensing, and light manipulation in environments characterized by intense scattering.

This section elaborates on the theoretical framework developed to describe the spectral memory effect in scattering media, integrating concepts of random matrix theory and diffusion theory. This model is established on the assumptions of Mie scattering and the illumination of coherent light. We denote the scattering matrix as T={Tm,n}(1≤m,n≤N), where N represents the number of spatial modes at both the input and output planes. In Cartesian coordinates, each element of this matrix reflects the transformation of the field from a specific point on the input plane to another on the output plane. According to diffusion theory, light can follow various paths between two points with probability density function P(L) of paths’ lengths L, which is distributed according to statistical diffusion specified in scattering media as P(L)=c(4πDL)3/2exp(−r24DL−μact).39^,60 The Beer-Lambert law suggests that the matrix element for a given wavelength λ is the cumulative effect of these paths: Tm,n(λ)=∑pe(−α+ik)Lp.(4)Here, α represents the absorption coefficient, k is the wavenumber, and Lp signifies the length of the p’th path, a variable subject to randomness. Considering a small detuning in wavelength or wavenumber (Δλ or Δk), the evolution of the matrix element is given by $$\begin{gathered} Tm,n(λ+Δλ)=∑pe[−α+i(k+Δk)]Lp \\ =∑pe(−α+ik)Lp(1+iΔkLp). \end{gathered}$$(5)

In deriving Eq. (5), we assume that both L and α remain the same, applying a first-order Taylor expansion to account for the minor phase shift caused by Δk. This derivation describes how transmission characteristics shift with slight wavelength adjustments, forming our model’s basis. Given L statistical independence from the phase term e(−α+ik)Lp, we can simplify Eq. (5) to $$\begin{gathered} Tm,n(λ+Δλ)=∑pe(−α+ik)Lp+iΔkL¯χnorm∑pe(−α+ik)Lp \\ =Tm,n(λ)[1+χ(Δλ)]. \end{gathered}$$(6)

Here, L¯ denotes the average path length from a collection of Lp, and χnorm is normalized due to the spectral response, conforming to a complex-valued Gaussian distribution as per the central limit theorem. This adjustment, facilitated by a tunable spectrum, alters path length distributions to include any conceivable transmission response, eliminating independence among them. Accordingly, the wavelength-dependent scattering matrix is formulated as T(λ+Δλ)=βT(λ)[I+X(Δλ)].(7)Here, I is the identical matrix and β is a regularization coefficient for energy conservation. X(Δλ) represents the detuning matrix with elements following the distribution χ(Δλ), nearing unitary. The variance (standard deviation) of the Gaussian distribution can be succinctly expressed as σ2/N (σ/N), where the value of σ manifests its representation of the spectral detuning parameter concerning the Δλσ=L¯Δk=L¯2πΔλ/λ2.(8)

Equation (7) demonstrates modeling a minor wavelength detuning as a perturbation on the original scattering matrix, facilitating most calculations at the wavelength λ. This approach provides a robust foundation for understanding and computing the spectral memory effect within scattering media.

Drawing upon random matrix theory,45^,54 we decompose the scattering matrix of scattering media via singular value decomposition: T=UτV†. Here, U and V are unitary matrices, representing the transformations from the input plane to the eigenchannels and from the eigenchannels to the output planes, respectively. The matrix τ, diagonal in form, contains the eigenvalues of these channels in descending order. The operator † denotes the conjugate transpose of a matrix. To specifically excite the n’th eigenchannel, the scattering medium is illuminated with Vn (the n’th column of V), producing an output field En(λ) as follows: En(λ)=T(λ)Vn=τnUn,(9)where τn symbolizes the energy transmitted. When the illumination wavelength is detuned to λ+Δλ, the resultant output field En(λ+Δλ) can be computed as En(λ+Δλ)=T(λ+Δλ)Vn=βT(λ)[I+X(Δλ)]Vn.(10)

The correlation coefficient for the n’th eigenchannel, indicating the similarity between output intensity distributions for the base and detuned wavelengths, is defined as Cn(Δλ)=⟨|En†(λ)En(λ+Δλ)|2⟩⟨En†(λ)En(λ)En†(λ+Δλ)En(λ+Δλ⟩,(11)where the operator ⟨·⟩ denotes an ensemble average. The calculation for the numerator of Eq. (11) simplifies to $$\begin{gathered} Nominator=⟨|τnUn†βT(I+X)Vn|2⟩=⟨β2τn|Un†[T+TX]Vn|2⟩ \\ =β2τn(⟨Un†TVnVn†T†Un⟩+2⟨Un†TRe(VnVn†X†)T†Un⟩+⟨Un†TXVnVn†X†T†Un⟩) \\ =β2τn(τn+0+τnσ2/N) \\ =β2τn2(1+σ2/N), \end{gathered}$$(12)where XVnVn†X† derives the average induction norm for X(Δλ) as σ2/N. This outcome leverages time-reversal symmetry to relate back to Eq. (9), such that T†Un=τnVn. The cross-term is averaged out because X does not appear in pairs. For the denominator, we find $$\begin{gathered} Deominator=τn⟨β2Vn†[T(I+X)]†T(I+X)Vn⟩ \\ =β2τn(⟨Vn†T†TVn⟩+2⟨Vn†Re(X†T†T)Vn⟩+⟨Vn†X†T†TXVn⟩) \\ =β2τn(τn+0+σ2Tr(T†T)/N) \\ =β2τn(τn+τ¯σ2). \end{gathered}$$(13)

Here, τ¯=Tr(T†T)/N is the average transmission across all eigenchannels, and the operator Tr(·) is the trace of a matrix. The second term in Eq. (13) encapsulates the collective effect of interacting with the scattering media via a random wavefront of intensity σ2. Merging these findings, we derive the correlation coefficient for the n’th eigenchannel as Cn(Δλ)=τn(1+σ2/N)τn+τ¯σ2.(14)

The spectral detuning parameter σ2, integral to Eq. (14) and influenced by the detuning matrix X, inherently incorporates the detuning wavelength Δλ, serving as a direct measure of the spectral memory effect’s magnitude. This formula demonstrates the mouldability of the spectral memory effect via a strategic selection of scattering eigenchannels with varied transmission properties.

This section outlines the methodology for simulating the spectral memory effect in scattering media using the FDTD method. Our model employs a two-dimensional setup that incorporates numerous cylinders, randomly placed within a square enclosure, to mimic the scattering environment. These cylinders are represented by blue circles, with refractive indices nc=2 and diameters d varying between 150 and 500 nm, as illustrated in Fig. 6(a) for visualization. The surrounding space, or ambient geometry, has a refractive index nb=1, framed by an orange boundary with dimensions W×L=12μm×5μm. The FDTD mesh grid within this frame has a resolution of 10 nm. The chosen configuration, after statistical evaluation, presents a scattering coefficient μs=2.703mm−1 and an absorption coefficient μa=0, indicating no absorption. Single-frequency plane waves, depicted by red arrows and entering from the left at a wavelength of 640 nm, act as the angular probing sources. Scattering matrix characterization spans the angular spectrum ky along with the y-direction, with transverse magnetic waves polarized along the z-direction. A yellow box marks the field monitor area, capturing the angular spectrum of the electromagnetic wave over a full surveillance range from −π/4 to π/4. The simulation’s boundary, the outermost orange frame, is treated as a perfect matching layer to ensure perfect absorption, thus mimicking real-world scenarios where photons propagate into the far field without reflection. By individually adjusting the incident angles, angular transmission matrices are detailed, enabling wavefront control through the synthesis of angular plane waves.

For simulations incorporating absorption, the cylinders’ imaginary refractive index part is set to 0.001i, leading to an attenuation coefficient μa=0.019mm−1. To simulate finite bandwidth effects, the model substitutes the single-frequency wave with a broadband source spanning a 2-nm-bandwidth, achieved by scanning detuning sources in 0.5 nm steps. This comprehensive setup allows for a detailed examination of the spectral memory effect under various scattering and absorption scenarios.

Upon configuring the scattering setup, the FDTD simulation facilitates the characterization of the angular scattering matrix. By applying singular value decomposition to this matrix, different scattering eigenchannels can be selectively activated through precise combinations of angular illuminations. This interaction with the cylindrical scattering structure enables the determination of the spectral correlation relationships among scattering eigenchannels under varying conditions, including absorption and finite bandwidth models. Illustrated in Figs. 6(b) and 6(c), we present the correlation coefficients across an array of scattering eigenchannels for three distinct detuning wavelengths: 2, 4, and 6 nm. The data are analyzed using Eq. (14), producing fitting curves that align closely with the theoretical predictions from both standard simulations and experiments. By extracting the spectral detuning coefficients σ from these fitting functions, we can graphically demonstrate the impact of detuning wavelengths on the spectral memory effect. Our findings, detailed in the main manuscript, highlight how practical factors—namely, absorption and the finite bandwidth of the illumination—significantly influence the enhancement of the spectral memory effect. This comprehensive approach not only validates our proposed derivations but also elucidates the complex interplay between scattering properties and external conditions that affect the spectral memory effect.

During experiments, a resin sample was selected as the primary scattering medium. This section details the fabrication process of the resin samples. Epoxy resin, chosen for its clear appearance and fluidity at room temperature, serves as the foundation. To induce scattering, alumina particles, making up 1% of the mixture’s total mass, are incorporated. The mixing process involves a magnetic stirrer to achieve an even and homogeneous distribution of the alumina particles within the resin. Following this, an amine hardener is added at a ratio of ∼1/4 of the total mixture mass to initiate solidification. The mixture is stirred once more with the magnetic stirrer for several minutes to remove any air bubbles present. The final step involves transferring the mixture into a predetermined mold and curing it in a vacuum oven set at 80°C. This process solidifies the mixture, yielding a solid resin block that can be shaped as required. The result is a resin sample with embedded alumina particles, ready for use as a scattering medium in our experiments.

Biographies of the authors are not available.