Photon localization transition in a magnetorheological fluid

The phenomenon of electromagnetic (EM) wave propagation in disordered media¹ has garnered considerable interest in the realms of optics and condensed matter physics.^2,3 In such media, the interaction of multiple scattering events induces changes in wave amplitudes and phases, leading to mesoscopic correlations due to the overlapping of wave paths. Under conditions of weak disorder, waves traverse the medium diffusively, with minimal regard for the phases of the scattered waves. Conversely, in the presence of strong disorder, the phases of scattered waves become critically important, resulting in energy becoming localized at random locations—a phenomenon known as Anderson localization, which effectively impedes wave transport. This concept was initially introduced by Anderson in the context of electronic waves,⁴ but it has since been extended to various types of waves, including optical waves. Optical systems have served as invaluable experimental platforms for observing these transport behaviors, showcasing phenomena, such as diffusion, weak localization, and Anderson localization, thereby contributing significantly to the study of mesoscopic physics.^3,5–7

Magnetically tunable fluids, particularly magnetic nanofluids (MFs) and magnetorheological fluids (MRFs), offer unique platforms for investigating the effects of external magnetic fields on light propagation. MFs typically consist of nanometer-sized magnetite particles suspended in a carrier fluid, while MRFs constitute a bidispersion of larger micrometer-sized magnetizable spheres suspended in a magnetic nanofluid. The ability to manipulate the arrangement and dynamics of these particles using an external magnetic field makes MFs and MRFs, especially intriguing for studies of EM wave transport. While the properties of transmitted light in MFs have been explored in previous studies,^8–15 there is a paucity of literature focusing on the statistical dynamics within MRFs. Studies have demonstrated that these fluids can exhibit dramatic changes in their optical properties in response to magnetic fields, resulting in intriguing effects, such as the inhibition of transmitted light and enhanced light–matter interactions. The works of Mehta et al.^9,11,16,17 revealed that a nanosoup concoction made from MRF leads to the extinction of total transmission. Their work sparked debate among critics about whether their system provided an experimental demonstration of Kerker’s theory on scattering due to small magnetic particles, which leads to the extinction of transmitted light, or if the effect was due to absorption or actual light localization within the system. Nonetheless, there is consensus that this intriguing system based on bidispersed magnetic fluids is an interesting candidate for studying mesoscopic transport.^18,19 Previous studies in magnetically tunable fluids have shown that a photonic bandgap (PBG) can be magnetically induced and tuned in a colloidal dispersion of magnetic spheres, potentially useful for lab-on-chip devices^20,21 and how a magnetic field can induce spatial order in a ferrofluid, creating a tunable system that exhibits characteristics of both Anderson localization and Bragg scattering.²² In particular, an interesting idea of lossy Anderson–Bragg cavities was proposed based on theoretical analysis of a dissipative disordered system that emulates the ferrofluid.²² Although this theory makes a direct reference to Anderson localization in ferrofluids, to our knowledge, there have been no direct experimental follow-ups that provide experimental data implying the possibility of localization. This is understandable given the fact that experimental characterization of Anderson localization is not trivial, and specialized mesoscopic quantifiers have been created for the same. To our belief, the lack of experimental characterization of transport in such systems leaves a major gap in the physics of mesoscopic transport.

Motivated by this gap, our study systematically investigates the propensity of photon localization in magnetically tunable fluids. By examining the behavior of both magnetic nanofluids and magnetorheological fluids under varying magnetic field strengths, we want to assess the probability of localization in these complex fluids. By carefully controlling the system parameters, we want to observe the dynamic changes in the optical scattering properties of the fluids and study how different magnetic field strengths influence the arrangement and dynamics of the particles within the fluids and how these changes would impact the propagation of light through the medium. Our comprehensive approach allows us to quantify key parameters that are typically used to identify photon localization in such samples with absorption and to develop a deeper understanding of how external magnetic fields can be used to control light propagation in disordered media.

We employed two distinct types of multiple scattering media for our experimental investigations: (1) a magnetic nanofluid (MF),¹¹ which consists of a kerosene-based suspension of magnetite (⁠ $Fe 3 O 4$⁠) nanoparticles with an average size of 18 nm, coated with oleic acid, and dispersed in kerosene and (2) a magnetorheological fluid (MRF),¹⁷ which comprises 1  $μ$m magnetite spheres, also coated with oleic acid, dispersed within the kerosene-based MF. The particle concentrations in both media were meticulously adjusted to maintain transparency with an optical path length of approximately 2 mm. For our light source, we utilized a linearly polarized He–Ne laser operating at 632 nm with a power output of 7 mW. As illustrated in Fig. 1(a), the laser beam underwent spatial filtering (SF setup) and was directed by a mirror (M) into the sample (S), which was housed in a quartz cuvette with dimensions $20×2×50$ mm $3$ and positioned within a Helmholtz coil (HC) setup. This HC setup was capable of generating a uniform magnetic field up to 800 G, oriented perpendicular to the direction of the incoming light. We collected the transmitted light using a 4f imaging setup consisting of two lenses with focal lengths of 10 cm each (L3 and L4), and a CCD camera was used to measure the scattered light along the transmission path. At the object plane of the 4f setup was the speckle pattern at some distance from the cuvette, which was reproduced on the CCD.

The scattering profile and its time-dependent changes under the influence of a magnetic field were key aspects of our study. Figure 1(b) demonstrates how the scattering profile evolves over time when a constant magnetic field strength of 650 G is applied to the MF. As observed in prior studies,^8,9,12,13,16 the transmitted intensity decreases progressively with increasing magnetic field strength. This observed trend in transmitted intensity is primarily due to the magnetic field-induced formation of chains within the fluid, which affects the scattering characteristics. The specific behavior of the scattering profile over time provides insights into the dynamic processes within the fluid. For the MF case, detailed in Fig. 1(c), a stable transmitted intensity is recorded at a magnetic field strength of 300 G (gray), indicating that this field strength is not sufficient to induce the formation of chains. However, when the magnetic field strength is increased to 450 G (orange), the transmitted intensity begins to decrease, indicating the onset of chain formation. At higher magnetic field strengths, such as 525 (wine) and 650 G (black), the transmitted intensity drops sharply and then stabilizes after approximately 80 s. This stabilization occurs because the length of the field-induced chains becomes constant over time, and the system reaches an equilibrium state (see  Appendix A). In contrast, for the MRF, the system’s evolution under the magnetic field occurs at lower magnetic field strengths [Fig. 1(d)]. A stable transmitted intensity is observed at 75 G (gray) where no chain formation has taken place yet. The typical intensity drop and stabilizing behavior is observed for 400 (orange), 500 (wine), and 550 G (blue). The interesting thing to notice in the case of MRF is that the larger magnetite spheres result in a slower system evolution, with equilibrium being reached after approximately 140 s at a field strength of 550 G, as shown by the blue curve in Fig. 1(d). The dynamics of the systems’ evolution are further influenced by the applied magnetic field strength. The rate of change increases with stronger magnetic fields in both cases, as indicated by the subsequent gray, orange, wine, and blue curves in Figs. 1(c) and 1(d). The changes in transmitted intensity at different magnetic field strengths provide a comprehensive view of the behavior of these fluids. As the scatterer size increases and the number density decreases, the transmitted intensity in both MF and MRF begins to rise again until the system reaches equilibrium. This phenomenon is more pronounced in MRF due to the larger size of the scatterers in addition to the MF. At high magnetic field strengths, the transmitted intensity in MF dropped from an initial value of 1 to 0.6, whereas for MRF, it exhibited a significant reduction to 0.25 from its original intensity of 1. To further understand the reasons behind such a reduction in transmitted intensity, we conducted a detailed study of the evolution of intensity statistics.

The phenomenon of light localization, characterized by a decrease in transmitted intensity due to wave trapping within the scattering medium, requires us to account for all potential causes of light extinction. First, we addressed absorption by ensuring that the input wavelength (632 nm) did not fall within the peak absorption spectrum of the magnetic nanofluid (details provided in  Appendix B). In this work, we utilized a narrowband source at 632 nm, where the absorption is approximately 25% of the peak absorption. Nonetheless, the generalized conductance parameter (⁠ $g ′$⁠) used in our analysis effectively accounts for both scattering and absorption effects, allowing us to reliably quantify light transport. Second, we considered the effect of the magnetic field on the scatterer size within the sample. In the magnetic nanofluid, nanoparticles tend to form chains due to head-on aggregation along the direction of the applied magnetic field, driven by field-induced dipolar interactions. The observed decrease in transmitted intensity for MF can be explained by Kerker’s theory^23,24 in the small particle limit, where forward scattering becomes zero when the scatterers satisfy the condition $μ=1/ε$⁠, with $ε$ being the relative electric permittivity and $μ$ the relative magnetic permeability. In MRF, the chains formed are longer and thicker, involving more particles compared to those in MF, resulting in different dipolar interactions within and between the chains.^25–27 Intensity statistics is an important parameter in revealing insights into wave localization phenomena and the fundamental principles governing light propagation in disordered media. These metrics are crucial for understanding the crossover from exponential to diffusive scaling of transmission.^28–33 In the field of ferrofluids, it has been shown that a magnetic field applied to magnetic fluids induces a photonic bandgap (PBG) at optical frequencies, primarily through the formation of linear arrays of nanomagnetic particles acting as Bragg gratings.^20–22 This magnetic field-induced structuring causes multiple scattering, leading to the creation of a PBG. Our observations suggest that the magnetically induced PBG occurs around 525–550 G for the bidispersion at 632 nm (with dissipation). The interplay between disorder and the PBG becomes evident as the system evolves. However, one should note that in this work, the term “photonic bandgap” (PBG) is used in a broader sense to describe the magnetically induced bandgap that arises due to the field-induced arrangement of magnetic particles. While this bandgap shares certain characteristics with a traditional PBG, such as reduced transmission and altered light propagation, it is more accurately described as a “magnetically induced bandgap,” as seen in earlier studies on magnetically tunable fluids.^20–22 Previously, our group has experimentally characterized intentionally disordered photonic crystals, in a non-Hermitian environment, with $P( I/⟨ I⟩$⁠) and the generalized conductance $g ′$⁠, to marking the transition from delocalizing to localizing disorder.³⁴ We apply the same analysis to the current system to identify the domain of light transport. We note that the robustness of our findings at 632 nm, combined with the use of generalized conductance, enables us to capture the transport dynamics even in the presence of absorption. While one could carry out experiments at higher wavelengths to bypass absorption, we note that higher wavelengths also experience weaker scattering. Thus, we expect a lower propensity of $g ′$ reaching values close to 1.

To analyze the intensity statistics of changing speckle patterns, we exposed our system to high magnetic field strengths. In transmission studies involving scattering experiments, it is essential to eliminate the contribution of ballistic light to accurately analyze the scattered light. Several techniques exist for this purpose. For instance, in pulsed laser experiments, this can be achieved through time gating, which selectively filters out the first-arriving ballistic photons. In the case of continuous-wave lasers, Fourier filtering methods, such as a spatial filter-beam blocker combination, are employed to block the direct transmission of ballistic light. In our study, we specifically remove the ballistic beam from the received transmitted intensity pattern. See  Appendix C for details. We fit a 2D Gaussian to the ballistic spot and select a region located at least two times the Full Width at Half Maximum (FWHM) away from the Gaussian center. This approach ensures that only the multiply scattered light is considered in the final analysis. It is important to note that the statistical measurements remain consistent regardless of the chosen area, as long as it is away from the ballistic spot. Figures 2(a), 2(b), and 2(c) illustrate the speckle map of MF when exposed to 650 G field strength at time instances of 5, 50, and 100 s. The intensity distribution of the speckle patterns in the case of MF at two magnetic field strengths, 525 (wine) and 650 G (blue) applied for 5, 50, and 100 s, respectively, are shown in Figs. 2(d), 2(e), and 2(f), where the gray line shows the Rayleigh fit. We observe that the distribution remains sub-Rayleigh^35,36 for all times.

Theoretically, the intensity fluctuation distribution follows a precise analytical behavior, which allows for a best-fit determination of the conductance, $g$⁠. However, given that we are dealing with an absorbing system, to examine conductance fluctuations, we evaluated the generalized conductance $g ′$ for each configuration as the system evolved with time. Defining $S a b$ as the total transmission normalized to the ensemble average (⁠ $S a b= T a b/⟨ T a b⟩$⁠), $g ′$ is calculated as $4/3[var( S a b)−]$⁠.²⁸ A condition where $g ′<1$ indicates localization. This method has been utilized previously to use the variance of measured intensity distributions to compute $g ′$ across a broad range of disorder configurations.^31,34,37 In MF, the generalized conductance [Fig. 2(g)] shows a decreasing trend at high magnetic field exposure of 525 (wine) and 650 G (blue), but the value always remains greater than 1, indicating the system remains in the weak scattering regime. In the case of MRF, the intensity distribution shows significant transition as the system evolves over time with the application of higher magnetic field strength. Figures 3(a), 3(b), and 3(c) illustrate the speckle evolution at times 5, 90, and 130 s with exposure to 525 G field strength. Figures 3(d), 3(e), and 3(f) show the intensity distribution at two strong magnetic field strengths, 525 (wine) and 550 G (blue) at three time intervals: 5, 90, and 130 s. The gray dashed line shows the Rayleigh fit. We observe that, with increasing time interval, the distribution tail begins to stretch, which is a classic indicator of a mesoscopic system transitioning into the localizing regime. Consequently, the generalized conductance value [Fig. 3(g)] also falls below 1 as the system attains equilibrium around the same time interval, which is also commensurate with the transition from a delocalized to localizing domain.²⁹ Further, unlike in the nanofluid, we observe an initial dip followed by a peak in the generalized conductance for MRF at around 40 s. This difference likely arises from the distinct field-induced dynamics in MF and MRF. Given the obvious difference in the two sizes of the magnetic particles in the bidispersion, one can expect differential response timescales to applied magnetic fields. Specifically, the larger micrometer-sized particles may experience a delay in the formation of complex thick chains, with the nanoparticles aggregating slightly earlier. The dip and rise behavior in generalized conductance may indicate this staggered aggregation process. However, rigorous analysis of this peak is not the focus of this work. Rather, we analyzed the statistics of the generalized conductance at magnetic field strengths of 525 and 550 G, where the symptoms of incipient localization were evident. We consider the generalized conductance values for the time around which the system transitions into a localizing regime (75 s onward) as illustrated in Fig. 4(a). Figures 4(b) and 4(c) present the generalized conductance distribution at these field strengths, showing a distinct peak for $g ′≤1$ at 525 and 550 G, respectively. In the current scenario, we speculate that our system assumes disorder configurations sufficiently strong to sustain incipient localization.

Additionally, we measured the temporal cross correlation of the field-induced speckles, as depicted in Figs. 4(d) and 4(e). Initially, the speckles exhibit high correlation, which decreases over time, reaching minimum correlation values of 0.06 and 0.09, respectively. By fitting the correlation widths to Gaussian distributions and plotting these widths against time, [Fig. 4(f)], we assessed the duration of sustained correlation within the system. A minimum correlation width corresponds to the lowest intensity at each magnetic field strength, while the maximum correlation width is observed at equilibrium. Furthermore, the cross-correlation distribution during the transition to the localized regime was plotted and fitted with a Gaussian distribution (blue dashed line), as shown in Figs. 4(g) and 4(h) signifying the decorrelation arising due to system evolution.

Through our investigation, we identified and analyzed the propensity of photon localization in magnetically tunable fluids, focusing on magnetic nanofluids (MF) and magnetorheological fluids (MRF). The results distinctly demonstrate differences in EM wave scattering behavior between these two types of fluids when subjected to varying magnetic field strengths. Magnetic nanofluids exhibited weak scattering without significant localization effects across all field strengths, indicating that the magnetic field does not sufficiently alter the nanoparticle arrangement to induce wave trapping. In contrast, magnetorheological fluids showed a strong propensity of localized light transport with increasing magnetic field strength. This shift was evidenced by reductions in transmitted intensity, alterations in speckle patterns, and significant changes in intensity statistics, particularly the generalized conductance parameter $g ′$⁠. Specifically, at field strengths of 525 and 550 G, we observed peaks for $g ′<1$⁠, indicating possible incipient localization. Temporal cross-correlation measurements of field-induced speckles revealed an initial high correlation that decreased over time, with minimum values of 0.06 and 0.09. By fitting correlation widths to Gaussian distributions and plotting them over time, we determined that the duration of sustained correlation increased as the system reached equilibrium. Furthermore, the formation of particle chains in MRFs under magnetic fields plays a significant role in influencing optical properties and facilitating multiple scattering. The effect is more pronounced in MRFs due to the larger particle size and varied structural geometry. Our statistical studies highlight that the indicators of localization are associated with the magnetic-field-induced photonic bandgap in MRFs.

These findings signify the importance of particle size, density, and magnetic field strength in tuning the optical properties of magnetorheological fluids. The insights gained from this study have potential applications in optical switching, adaptive optics, and the design of smart materials through controlled light manipulation using external magnetic fields. This work enhances our comprehension of light behavior in complex fluids under magnetic influence, offering valuable contributions to the understanding of mesoscopic transport phenomena and the development of advanced photonic materials.

We thank Krishna Joshi and Rounak Chatterjee for useful discussions and helpful comments. We acknowledge funding from the Department of Atomic Energy, Government of India (Grant Nos. 12-R and D-TFR-5.02-0200).

The authors have no conflicts to disclose.

Himadri Sahoo: Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Writing – original draft (lead). Kinnari Parekh: Project administration (equal); Resources (equal); Writing – review & editing (equal). Junaid Masud Laskar: Project administration (equal); Supervision (equal); Writing – review & editing (equal). Sushil Mujumdar: Conceptualization (lead); Funding acquisition (lead); Project administration (lead); Resources (equal); Supervision (lead); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

We examined the size distribution of the chain lengths of aggregates formed when exposed to a constant magnetic field. This was done by incorporating an objective based imaging via illuminating the magnetic nanofluid with white light and then capturing the chain formations using CCD. Figures 5(a), 5(b), and 5(c) illustrate the size distribution of magnetic field-induced chains at times 5, 50, and 100 s when exposed to a constant magnetic field of 375 G. We observe that as the time progresses from 5 to 50 to 100 s, the counts (y axes) drop from $10 4$ to $10 3$ to $10 2$ indicating the change in number density with an increase in the field-induced chain length. Consecutively, the distribution becomes uniform when we approach 100 s indicating that the system attains equilibrium in terms of the maximum chain length. The corresponding distribution of chain diameters is shown in Figs. 5(d), 5(e), and 5(f) at the same time intervals. We observe that there is no significant change in the diameter value. The average chain length and diameters over time are shown in Figs. 5(g) and 5(h). The average chain length further confirms that beyond 80 s, the system attains equilibrium and the chain length becomes constant.

The absorption cross section, $σ$⁠, is a measure of the probability of an absorption process occurring when a material is exposed to EM radiation. It quantifies how effectively a material or particle can absorb light at a given wavelength and is typically expressed in units of area.

We computed the absorption cross section across various wavelengths for magnetic nanofluid with a concentration of 4.97 mol l $− 1$ by calculating the molar extinction coefficient from Eq. (B1) and incorporating in Eq. (B2), as shown in  Appendix B. Figure 6 illustrates the absorption cross section vs wavelength. For $λ=632$ nm, the absorption cross section $σ=1.834× 10 − 24$ cm $2$⁠.

In order to specify the speckle area for computing the intensity statistics, it was crucial to ensure that the chosen region (denoted by the white dashed rectangle) was positioned far from the ballistic spot. This precaution was necessary to avoid any influence of the ballistic spot on the intensity statistics. To achieve this, a 2D Gaussian surface fit was performed on the ballistic spot at 0 G. By fitting a 2D Gaussian function to the spot, we were able to accurately determine its Full Width at Half Maximum (FWHM) along the x axis (FWHMx) and y axis (FWHMy). These measurements provided a precise characterization of the ballistic spot, enabling us to confidently select a speckle area that was sufficiently distant from the spot for reliable intensity analysis (Fig. 7).

The parameters obtained from the fit are listed in Table I.

From the above fit, we chose an area two times FWHMy to conduct our speckle analysis ensuring the statistics are not influenced by the ballistic light.