# Observation of the photonic Hall effect and photonic magnetoresistance in random lasers

### Theoretical framework

As shown in Fig. 1, the pump laser is coupled into one end of the magnetic gain polymer optical fiber (MGPOF), the generated RL and the residual pump laser exit from the other end of MGPOF, which are oriented along the x-axis. The direction of the applied magnetic field is along the z-axis. The RL is formed by the disordered multiple scattering of photons in the gain medium. From the macroscopic experimental phenomenon, the applied magnetic field causes the RL to produce magnetic transverse photocurrent *J*, which proves the PHE in the RL system. Moreover, the intensity of the RL emission is weakened by the applied magnetic field, which proves the PMR in the RL system. From the microscopic disordered multiple scattering during RL generation, we note that contrary to the commonly assumed view that nanoparticles have a constant global magnetic moment, in fact, the applied magnetic field can significantly increase the magnetic moment of ferrite nanoparticles. The magnetization volume of nanoparticles is closely related to the surface disorder of their structure, and when a magnetic field is applied, the disordered surface spin is gradually polarized, resulting in an increase of more than 20% in the magnetic volume^{37}. In ref. ^{37}, the magnetic scattering amplitude of small angle neutron scattering (SANS) is used to determine the morphology of magnetic nanoparticles. It is found that the magnetic core radius *r*_{mag} increases with the increase of the external magnetic field strength. The variation of the severe magnetic scattering amplitude with the applied magnetic field is simulated based on the micromagnetic theory. The simulation results show that the fluctuation of magnetic parameters, that is, the contribution of magnetocrystalline anisotropy and magnetostriction, is the most likely source of the variation of magnetic radius with magnetic field. The magnetic volume and the corresponding magnetic field energy increase with the increase of the applied magnetic field strength are obtained by calculating the Zeeman free energy. The core-shell Fe_{3}O_{4}@SiO_{2} NPs used in this work and the CoFe_{2}O_{4} NPs sphere in (37) surrounded by the oleic acid ligand layer behave similarly. That is, the size of the magnetized core increases with the applied magnetic field because they have similar high surface-to-volume ratios, similar surface atom distribution, and similar magnetic resonance lines in the range 0–800 mT^{38,39,40}.

As shown in the pink box of Fig. 1. The upper semicircle represents the structural morphology, and the lower semicircle represents the magnetic morphology. Prior to the application of a magnetic field, the structural and magnetic morphology of nanoparticles are equal in size, as shown in the circle on the left, whereas the initially disordered surface spin is gradually polarized in the applied magnetic field, causing the magnetic radius to increase beyond the structurally disordered surface region as shown in the circle on the right. The deep pink squares show the structurally coherent grain of the Fe_{3}O_{4} core, and the light pink dots show the structural disorder of the Fe_{3}O_{4} surface region. Note that the SiO_{2} shell is not shown in the morphology of ferrite NPs. The silica shell on each NPs has its amorphous nature. The white arrows represent the collinear magnetic dipole spin in the magnetic core, and the purple arrows represent the spin disorder of Fe_{3}O_{4} surface region. The overall magnetic moment of the core-shell structure Fe_{3}O_{4}@SiO_{2} magnetic NPs increases with the magnetic field. Following the Faraday effect, the magnetic field causes the rotation of the scattering polarization plane, which is linearly proportional to the component of the magnetic field toward the direction of light wave propagation. An increase in the applied magnetic field leads to an enhanced effect on the orientation of scattered light, causing an increase in the radius of the ordered magnetic dipole moment. This results in a weakened scattering disorder of RL, which ultimately reduces the RL intensity fluctuations.

In disordered multiple scattering, the anisotropy of the scattering cross-section is determined by the “anisotropy factor” *<cosθ*>, which is the average of *cosθ* divided by the phase function^{41}. *θ* is the angle of scattering deflection. In magnetic multiple scattering, *<cosθ>* can separate “up” scattering from “down” scattering. If the magnetic field is perpendicular to the incoming and outgoing wave vectors, there is a difference between the upward and downward flux along the magneto-transverse direction **k** × **B**. The magnetic anisotropy *η*(*η* ≠ 0) of a single scatterer in magnetic multiple scattering can be precisely quantified as the normalized difference between the total upward flux and the total downward flux as:

$$\eta \equiv 2\pi {\int }_{0}^{\pi }{{{{{\rm{d}}}}}}\theta \, {\sin }^{3}\theta {F}_{1}(\theta )$$

(1)

For a Rayleigh scatterer, the Born approximation can be used, which leads to *F*_{1}*(θ)* ∼ (*V*/*k*) *cosθ*^{42}. Multiple scattering of light in a disordered medium can be described by a diffusion tensor *D*, and Fick’s diffusion law relates the photon current density *J* defined per unit area to the local photon density gradient ∇*ρ*:

$$J=-D \, \bullet \, \nabla \rho$$

(2)

The magnetically induced off-diagonal component of the diffusion tensor *D* generates the magnetic transverse current density *△J*_{⊥}, which propagates perpendicular to the magnetic field **B** and the photon gradient ∇*ρ* and is mathematically denoted by \(\Delta {J}_{\perp }\propto {l}_{\perp }\hat{B}\propto \nabla \rho\). The gradient ∇*ρ* is the result of the diffuse propagation of photons in the scattering medium and is not parallel to the incident beam. The parameter *l*_{⊥} represents the coupling of photons within the scattering medium to the magnetic field. It is linearly proportional to the magnetic field and the Verdet constant *V*, and its dimension is the length. The difference between the upward and downward scattering magnetically induced photon currents is *△I*_{⊥}, △*I*_{⊥} = *I*_{U}–*I*_{D}, which is the integral of the magnetic transverse current density *△J*_{⊥} over the area of the photoelectric detector.

According to ref. ^{43}, *D* is expressed as

$${D}_{ij}({{{{{\bf{B}}}}}})={D}_{0}{\delta }_{ij}+{D}_{H}{\varepsilon }_{ijk}{B}_{k}+\Delta {D}_{\perp }({B}^{2}{\delta }_{ij}-{B}_{i}{B}_{j})+\Delta {D}_{\parallel }{B}_{i}{B}_{j}$$

(3)

In the above equation, the first term represents the typical anisotropic scattering, the second term represents the magnetic transverse scattering that produces PHE, and the third and fourth terms represent the reluctance terms that produce PMR. For PMR, the relative transmittance decreases as the magnetic field increases, and we need to focus on whether the emission intensity of the RL decreases with the applied magnetic field. We additionally analyze in Supplementary Material III the emergence of the PMR in RLs in terms of absorption due to the imaginary part of the dielectric constant. Nevertheless, due to the presence of inherent intensity fluctuations, we need to exclude in the RL regime whether the intensity of the RL happens to decrease in a fluctuating trend when no magnetic field is added.

On the other hand, the presence of an external static magnetic field can eliminate the contribution of a single scattering event to the overall multiple scattering process^{44}. As the magnetic field increases, the total scattering cross-section decreases^{45}. During laser irradiation in a multiple-scattering media, the scattering intensity is caused by the interference between light from different scattering paths. Two components are essential: the individual scattering events, for example, characterized by the radiation pattern of each scatter, and the light transmission in the medium due to all the scattering events. In the presence of a magnetic field, both components are affected. For the propagation, this is the well-known Faraday effect (and Voigt or Cotton-Mouton effect), described by the correction of the complex refractive index by the magnetic field^{1}. The imaginary part of the antisymmetric part of the dielectric constant can be interpreted as the Hall conductivity at optical frequencies when the multiple scattering in a disordered dielectric medium is exposed to an external magnetic field. In optical language, it mimics the “antisymmetric” extinction of the scattering mean free path. The real part of the antisymmetric term of the dielectric constant produces Faraday rotation and suppresses coherent scattering. The scattering cross-section measures the probability of occurrence of the scattering process, and a decrease in the scattering cross-section implies a decrease in the number of scattering events. In the disordered scattering process, since each single scattering event is associated with the disordered features of the RL medium, the reduction of scattering events also decreases the overall scattering disorder of the RL system. The scattering mean free path *ls* of RLs can be expressed as^{46}

$${l}_{{{{{{\rm{s}}}}}}}(\omega )=\frac{1}{\rho {\sigma }_{sc}(\omega )}$$

(4)

where *ρ* is the density of scattering nanoparticles and *σ*_{sc} is the scattering cross-section of a single nanoparticle with respect to frequency *ω*. So the decrease of the scattering cross-section *σ*_{sc} increases the scattering mean free path *l*_{s} of RL.

An important aspect to emphasize here is that disordered multiple scattering is essential to photon diffusion in RLs. Indeed, when the scattering direction of a single scattering particle is laterally deflected, the overall RL scattering disorder is reduced. Thus, proving the scattering disorder reduction is an important aspect for probing the existence of PHE in an RL. It is actually very difficult to directly observe the scattering disorder of the RL, though it can be experimentally inferred from the emission and fluctuation of the RL spectrum. Here we analyze the degree of disorder in the random scattering from its connection with the RSB approach to the photonic glassy phase above the RL threshold.

The transition from the photonic paramagnetic to spin-glass phase with RSB arises as the result of the interplay between the randomness and nonlinear couplings of the RL modes. The statistical characterization of both regimes is performed through the analysis of the so-called Parisi’s overlap parameter that essentially measures how intensity fluctuations of replicas of the RL system are correlated^{31}. As previously noted, RL spectra exhibit random fluctuations across a range of wavelengths. Our focus is on assessing the random intensity fluctuations that exist between pulses in RLs. The spectral intensity fluctuations at a certain frequency, \({\Delta }_{\alpha }(k)={I}_{\alpha }(k)-\bar{I}(k)\), are used to characterize the overlap between the intensity undulations of different replicas, where *I*_{a}*(k)* is the intensity of the light mode with replica index *α* and data collection index *k*, \(\bar{I}(k)\) is the average over replicas of the intensity of each light mode,

$$\bar{I}(k)=\frac{1}{N}{\sum }_{\alpha=1}^{{N}_{s}}{I}_{\alpha }(k),$$

(5)

*N* is the number of replicas, and Parisi’s replica overlap parameter^{31} is defined as

$${q}_{\alpha \beta }=\frac{\mathop{\sum }\nolimits_{K=1}^{N}{\Delta }_{\alpha }(k){\Delta }_{\beta }(k)}{\sqrt{\mathop{\sum }\nolimits_{K=1}^{N}{\Delta }_{\alpha }^{2}(k)}\sqrt{\mathop{\sum }\nolimits_{K=1}^{N}{\Delta }_{\beta }^{2}(k)}} .$$

(6)

Based on the measured spectra, a total of *N(N-*1*)/*2 values of *q* at each pumping energy are calculated to build the statistical distribution function *P*(*q*) of *q*_{αβ} values. Below the threshold, all RL modes oscillate independently during passive state operation, which corresponds to the paramagnetic state. Conversely, above the threshold, synchronous mode oscillation is impeded due to disorder, resulting in the photonic spin-glass phase. At low pump energies, the distribution *P*(*q*) of the overlap parameter *q* concentrates around zero with a Gaussian-like form, and the uncorrelated modes essentially do not interact in the paramagnetic phase. In contrast, above the RL threshold parameter *q* distributes in the range (−1, 1), with the emergence of two side maxima in *P(q)* around *q* = ±1, characterizing an RSB glassy regime in which the coherent oscillation of RL modes is hampered, and intensity fluctuations in distinct replicas are either correlated or anticorrelated. Here we count the sum of the extreme values of the distribution *P(q)* when *q*_{ab} is positive and negative as

$$P{(q)}_{\max }=P{(q)}_{\max+}+P{(q)}_{\max -}$$

(7)

The RL threshold represents the critical point between the photonic paramagnetic and glassy phases, also signaling the appearance of the RSB phenomenon.

### Observation of PMR in RL and the field-dependent RL spectroscopy

Spectra of RLs are presented in Fig. 2 and S4 for various magnetic field strengths and pump energies. Specifically, Fig. 2a displays the RL spectra for pump energies of 25 μJ while Fig. S4 displays the spectra for pump energies of 10, 15, 20, 30, 50, and 100 μJ. The measurements were taken at magnetic field strengths of 0, 50, 100, 200, and 300 mT for each pump energy. The spectrometer control software was used to collect RL spectra every 100 ms, with 2000 continuous measurements at different pump energy and magnetic field strength combinations. The RL spectra are generated by averaging the 2000 spectra. Results indicate that when no magnetic field is present (i.e., the magnetic field strength is 0 mT), the spectra transition from amplified spontaneous emission (ASE) at 10 μJ to spikes at 15 μJ and then to peaks at 20 μJ with increasing pump energy, demonstrating typical RL emission. As the pump energy continues to increase, the position of the RL peaks changes due to the disorder of RL scattering. This variation is caused by the disordered scattering of the RL. When the pump light enters the fiber, the laser dye provides gain, and the disordered multiple scattering of light between magnetic nanoparticles provides feedback. The nonlinear effects of saturation and mode competition will affect the performance of the pump^{47}. As the pump energy increases, the threshold of different modes will be reached, and new modes appear. General nonlinear theory is based on a self-consistent equation that determines how many modes exist in a fixed pump energy and the frequencies of these modes.

We observe in Fig. 2 that the intensity of both ASE and RL regimes decreases as the magnetic field strength increases in the seven sets of pump energies tested. Figure 2b displays the decreasing trend in peak intensity of the RL at each pump energy with increasing magnetic field strength, which shows the PMR effect in RLs. Near threshold, the RL intensity decreases by ~70% when the applied magnetic field is 300 mT. At sub-threshold and suprathreshold, the RL intensity decreases by >30%, and when the pump energy is much higher than the threshold, the energy decreases by a degree of more than 60% again. This suggests that the phenomenon of PMR in the POF RL system is more pronounced near and far above the threshold. The intensities of RL at different pump energy and magnetic field intensities are depicted in Fig. S4g, with the inflection point of the fitted line corresponding to the threshold. The inset in Fig. S4g is the magnification of the inflection point. The increase of applied magnetic field results in an increased RL threshold as shown in Fig. 2c. The first explanation for the PMR effect in RLs is based on the magnetic dielectric effect, where the electrons in each Fe_{3}O_{4}@SiO_{2} NP within the MGPOF become polarized in the presence of a constant magnetic field, leading to an increase in their dielectric constants^{48,49}. We elaborate on this interpretation in Supplementary Information III.

In this experiment, we utilized a 635 nm continuous laser (Changchun New Industries Optoelectronics, MGL-V-532-2W) to investigate whether the output transmitted light intensity changed with the magnetic field when this laser was injected into the same fiber as described above. RL can only be produced under pulsed laser excitation at 532 nm but not under continuous laser excitation at 635 nm. As depicted in Fig. 2d, an increase in the magnetic field resulted in decreased transmitted light intensity, proving that an increase in the magnetic field enhances the absorption and makes the intensity of the emission light decrease. It is worth noting that due to the inherent randomness of the RL, the intensity of its spectrum and the position of the peak fluctuate randomly in the limit of the gain medium and scattering intensity. To demonstrate the magnetic field’s modulating effect on RL intensity, Fig. 2e displays the 1st, 401st, 801st, 1201st, and 1601st of 2000 sets of continuously collected RL spectra without magnetic field at 20 µJ pump energy. The RL intensity gradually decreases when the magnetic field increases, while the RL intensity fluctuates irregularly when all other experimental conditions are the same and only the magnetic field is removed. The erratic RL intensity fluctuations rule out the attenuation of RL intensity by time-induced bleaching of the laser dye, which supports the modulating effect of the magnetic field on RL intensity.

### Observation of PHE in RL

Experiment on direct observation of the PHE in RL is shown in Fig. 3a. The pump laser is coupled into one end of the MGPOF, the generated RL and the residual pump laser exit from the other end. The magnetic field generated by the magnetic field generator (*z* axis direction) is perpendicular to the MGPOF (x-axis direction), and the MGPOF is completely within the magnetic pole range of the magnetic field generator. RL is collected by two PDs outside the magnetic field area and at the upper and lower positions (*y* axis direction) of the exit end face of the fiber. Filters are placed in front of PD1 and PD2 to filter the pump light. PD1 and PD2 are connected to the lock-in amplifier after the differential circuit to detect the magnetic transverse photocurrent △*I*_{⊥} = *I*_{U} − *I*_{D}. *I*_{U} and *I*_{D} are the light current tested by PD1 and PD2, respectively. It is then normalized by the transverse scattering intensity *I*_{⊥} = *I*_{U} + *I*_{D}. The ratio △*I*_{⊥}/*I*_{⊥} is used to describe the magnitude of the magnetic transverse photocurrent, which plays the same role as the Hall angle in the magnetic transport of electrons. The results shown in Fig. 3b confirm the linear dependence of the magnetic transverse photocurrent generated by the PHE on the magnetic field strength in a RL system. It is worth noting that the Fe_{3}O_{4}@SiO_{2} magnetic nanoparticles are randomly distributed in the core of the MGPOF, so the MGPOF does not have perfect spatial symmetry. Therefore, a “net zero” lateral deflection of RL emission at any given point along the MGPOF should not be expected. When there is no applied magnetic field, △*I*_{⊥}/*I*_{⊥} ≠ 0. As the intensity of the applied magnetic field B increases, the magnetic transverse photocurrent increases. This linear dependence of the magnetic transverse photocurrent on the applied magnetic field shows the occurrence of PHE in the RL. From Eq. (1) we conclude that the degree of PHE in the RL based on MGPOF is the slope of the red fitting curve *η* = 1.099 × 10^{−9}/mT, which is the evidence of PHE in RL systems.

### Field-dependent RSB in RL

Figure 4 and Fig. S5 illustrate the distributions *P(q)* of the Parisi overlap parameter for the correlations of RL intensity fluctuations at various magnetic field strengths and pump energies. Specifically, Fig. 4 displays the distribution obtained from the RL spectra for pump energies of 10, 15 25, and 50 μJ at 0, 50, 100, 200 and 300 mT. The three columns in Fig. S5 are for pump energies of 20, 30, and 100 μJ, respectively. The overall distribution *P(q)* exhibits narrow Gaussian-like distributions when the pump energy is 10 μJ below the threshold. However, as the pump energy increases to just above the threshold value of 15 μJ, *P(q)* changes significantly towards the double-peaked profile that characterizes the photonic RSB glassy phase in the RL regime. At 25 μJ pumping, the extreme values of *P(q)* are smaller, and a Gaussian-like form takes place at very high pumping energies of 50 μJ and 100 μJ. For very high pumping energies, the degree of disorder within the POF is greatly diminished and the replica-symmetric regime is recovered. Hence the profile of *P(q)* returns to the Gaussian-like form^{36,50,51,52}.

The impact of the magnetic field on the statistical distribution of correlations between RL intensity fluctuations can also be investigated. Figure 5 illustrates the distributions *P(q)* under different magnetic field strengths, ranging from 0 to 300 mT at different pump energies. A Gaussian function is fit to each distribution to determine the standard deviation *σ* for 10, 30, 50, 100 μJ. Interestingly, with increasing magnetic field strength, the standard deviation of the Gaussian distribution is reduced by more than 80% at 30 and 50 μJ. The decrease in the standard deviation with increasing magnetic field strength indicates that the scattering disorder of RL intensity fluctuations is attenuated by the orienting effect of magnetic particles on the magnetic field. This indicates a significant magneto-optical regulation in this pumping regime of the RL system based on POFs. The third column of Fig. S5 show a similar situation of the standard deviation *σ* at pump energies of 100 μJ, which reduces by 55%. At the pumping energy of 15 μJ, *P*(*q*)_{max} gradually increases by about 75% as the magnetic field increases, while the dispersion of the overall distribution *P(q)* reduces with a higher magnetic field when the pump energy is marginally above the threshold. The increase in the magnetic field results in a reduction in the disorder effect on the correlation of RL intensity fluctuations thanks to the orientation of the magnetic particles brought about by the magnetic field, which weakens scattering disorder. A similar increase in *P*(*q*)_{max} is observed when the pump energy is 20 μJ and 25 μJ, which increases by around 162% and 21%. It is noteworthy that at 10 μJ, the distribution *P(q)* shifts toward *q* = 0 when compared to 15 μJ and 20 μJ. When the pumping energy is further increased to 30 μJ, *P(q)* returns to an approximate Gaussian distribution as in the second column of Fig. S5. All these phenomena indicate that the increase in the magnetic field enhances the orientation effect on the magnetic particles, which then weakens the scattering disorder.

We should remark that the energy threshold also increases with the field, as shown in Fig. 2c. So, the increase of *P*(*q*)^{max} with the magnetic field in Fig. 5 cannot be solely used to directly infer the efficiency of the RL, since the effect of the proximity of the threshold must be also taken into account in the analysis. To be more specific, consider, for example, the case with excitation energy 15 μJ in Fig. 4f–j. Due to the dependence of the threshold on the field, we notice in Fig. 4g for 300 mT that the input energy 15 μJ is much closer to the threshold than in Fig. 4f, in which no field is applied. It is thus justified that *P*(*q*)^{max} is larger in Fig. 4g than in Fig. 4f (with *P*(*q*) more concentrated around the extrema *q* = ±1 in Fig. 4g), as actually observed. The same reasoning also applies to higher magnetic fields in Fig. 4h–j with 15 μJ, and for other values of excitation energy as well.

These RSB findings are also consistent with the analysis of the correlation coefficient *r (λ1, λ2)* between intensity fluctuations of modes at distinct wavelengths in the same replicas at each pump energy and magnetic field intensity^{53,54,55} in Supplementary Information V. We also probe the findings and analysis of RSB transition and correlation coefficient with numerical simulation results based on scattering theory in Supplementary Information VI.