Introduction

The Faraday effect (FE), first observed in 1845, provided the earliest experimental evidence of the interaction between optical radiation and magnetism. In this phenomenon, the application of a static magnetic field induces circular birefringence in a material leading to a rotation of the polarization plane of a propagating optical beam. The discovery of the FE marked a foundational milestone in the development of magneto-optics, a field that remains actively investigated to date. A notable contemporary example is the all-optical helicity-dependent switching (AO-HDS) phenomenon, which has attracted considerable attention in recent years1,2,3,4.

In AO-HDS, ultrashort circularly polarized (CP) optical pulses are used to control the magnetization order parameter. The effect was demonstrated in the single- and multi- pulse domains5,6, and recently even under continuous wave (CW) illumination7. The experimental evidence showed that the effect depends on a variety of parameters including magnetic structure8,9, material composition10,11, and laser parameters12,13. Consequently, numerous thermal14,15,16,17, photomagnetic18,19, and optomagnetic13 mechanisms have been discovered, which originate from the optical electrical field. The thermal mechanisms involve the interaction between electrons, spins, and a phonon bath, and include the ultrafast demagnetization driven by electron heating and absorption of the laser radiation as explored by Kampfrath et al.20,21,22. In the photomagnetic and optomagnetic mechanisms, the optical light irradiation induces a magnetic transition. These mechanisms include, for example, the optical spin transfer torque23,24 and optical spin–orbit torque25,26,27,28 that generate spin polarized currents that exert a torque on the magnetization.

Among the coherent optomagnetic mechanisms, the inverse Faraday effect (IFE), was found to play a vital role in many studies29,30,31,32,33. In 1966, Pershan et al.34 developed a phenomenological formulation of the IFE, where a magnetic moment is induced by the optical electrical field in a nonlinear process. According to this theory, the same second-order magneto-optical susceptibility of the FE, ({\chi }_{NL}^{{(2)}), is responsible also for the IFE. Consequently, the induced magnetization in the IFE depends on the optical intensity, (I), specifically, on the intensity difference between the right and left CP (RCP and LCP, respectively) components of the beam,* IRCP*−* ILCP*34,35,36,37. Recently, we showed that beyond the well-established effects that originate from the optical electrical field, also the magnetic field can contribute to the IFE38. Namely, according to the Landau-Lifshitz-Gilbert (LLG) equation, a magnetic torque emerges from the Zeeman energy of a CP optical magnetic field. This process was characterized by an interaction strength parameter, (\eta), which was determined from the ratio between the optical cycle and Gilbert relaxation times according to (\eta =\alpha \gamma {H}_{opt}/{f}_{opt}) where ({f}_{opt}), ({H}_{opt}), and (\alpha) are the optical frequency, magnetic field amplitude, and the Gilbert damping, respectively, and (\gamma) is the gyromagnetic ratio. For typical experiments using femtosecond pulses at 800 nm1,13,19,39, (\eta) is in the range of (\sim {10}}{-4}). Overall, in comparison to the empirical data31,38, the calculated torque was found to be sizeable yet insufficient to solely account for the measured values, highlighting the primary contribution of the electric field20,21,22,23,24.

These calculations rely on two assumptions: A) The first is that the macrospin approximation pertains. Under this approximation, the magnetization, (\overrightarrow{M}), is spatially uniform. When this assumption breaks down, (\overrightarrow{M}) nucleates into a texture and the LLG equation applies to the local domains in which the optical torque is uniform over all spins. B) The second assumption is that the losses are transverse such that (\left|\overrightarrow{M}\right|) is preserved. In the LLG equation the transverse losses are manifested by (\alpha), representing a viscous dissipation mechanism. These losses depend on the interaction timescale. When the timescale of the dynamics become comparable to the spin–orbit coupling timescales, the transfer of spin angular momentum to the lattice might be affected4. Additionally, the losses are governed by the spin diffusion length, which depends on temperature40,41,42. Therefore, the optically induced thermal heating14,15,16,17 also influences the loss dynamics. In this case, the Landau-Lifshitz-Bloch (LLB) equation43 describes the dynamics more accurately, as explored by Korniienko et al.44. In the quantitative experimental study of the optically induced torque in Ref.31, which served as a benchmark for comparison with the outcome of the LLG model38, the applied optical fluence resulted in a maximum demagnetization of 4%. Namely, the assumption that (\left|\overrightarrow{M}\right|) is conserved was overall valid.

Here, we show the relevance of the optical magnetic field also to the FE. We start by showing the properties of the optically induced torque that reproduce behaviors seen empirically in the IFE. These include the linear dependence on the optical fluence5,6,13,26,37 and more generally on ({I}_{RCP}-{I}_{LCP})34,35,36,37, the dependence (independence) of the longitudinal (transverse) torque on (\alpha)31, and the build-up of the torque in the multi-pulse and CW regimes. These similarities suggest that the contribution of the optical magnetic field may also be relevant to the reciprocal, direct FE. In this case, we find that the externally applied static magnetic field breaks the symmetry between the interactions with LCP and RCP radiation. By considering the linear magnetic susceptibilities of RCP and LCP radiation stemming from the LLG equation in the highly off-resonant limit, we derive an analytical expression for the Verdet constant. We find it to be wavelength independent, accounting for 17.5% of the measured value for Terbium-Gallium-Garnet (TGG) at 800 nm37, and up to 75% at 1.3 µm45. Lastly, we show that the Verdet constants resulting from the LLG equation for the FE and IFE are fundamentally different. This result reproduces the well-known breakdown of reciprocity between the FE and IFE resulting from the electrical field when reaching the out-of-equilibrium ultrafast timescales32,37,46,47,48.

Results

Implications of the optical magnetic field to the IFE

To show the relevance of the magnetic component to the FE, we begin by characterizing the optically induced torque and show that it displays similarities with previous experimental reports on the IFE. We start by examining the effect of a single optical pulse on the macroscopic (\overrightarrow{M}) as illustrated in Fig. 1a. To this end, we numerically integrate the LLG equation in which the losses are incorporated in the Landau–Lifshitz form38,49:

$$\frac{{d\vec{M}}}{dt} = - \gamma^{\prime}\left( {\vec{M} \times \vec{H}_{opt} + \frac{\alpha }{{M_{s} }}\vec{M} \times \vec{M} \times \vec{H}_{opt} } \right).$$

(1)

Fig. 1

Dependence of the torque on the pulse parameters. (a) Schematic illustration of the normalized longitudinal and transverse torques ({T}_{z}) and ({T}_{y}), respectively, induced by the optical pulse. (b) (|{T}_{z}|) after the application of an RCP Gaussian magnetic pulse as a function of ({\eta }_{H}), which is varied by sweeping ({H}_{peak}) for each (\alpha =0.025, 0.05) and (\lambda =800, 640\text{ nm}) value. ({\tau }_{p}=540\text{ fsec}). Dashed lines correspond to quadratic fits. (c) Dependence of (|{T}_{z}|) on ({\tau }_{p}), under ({\eta }_{H}=2\times {10}^{{-4}). Dashed lines correspond to linear fits. (d) ({T}_{z}) as a function of (\phi), under ({\eta }_{H}=2\times 1{0}}{-4}) and ({\tau }_{p}=540\text{ fsec}). Dashed lines correspond to (-sin(\phi )) fits. Panels (c) and (d) follow the color code of (b).

Here, ({M}_{s}) is the saturation magnetization and ({\gamma }^{{\prime}=\gamma {\mu }_{0}/(1+{\alpha }}{2})), where ({\mu }_{0}) is the magnetic permeability. We apply a right circularly-polarized (RCP) Gaussian pulse of the form (\vec{H}_{opt} \left( t \right) = H_{peak} \left( {\begin{array}{*{20}c} {\cos \left( {2\pi f_{opt} t} \right)} \ {\cos \left( {2\pi f_{opt} t - \phi } \right)} \ 0 \ \end{array} } \right)e^{{ - \frac{{(t - t_{peak} )^{{2} }}{{2\tau_{p}}{2} }}}}) where (\phi =90^\circ). The full-width at half maximum (FWHM) of the intensity is (2\sqrt{ln(2)}{\tau }_{p}) and the peak amplitude ({H}_{peak}) is reached at ({t}_{peak}). Throughout the simulations, (\overrightarrow{M}) is initialized in (\widehat{x}) and ({M}_{s}=3\times 1{0}^{5} A/m) which is typical of Co-based films used experimentally31.

We first examine the dependence of the longitudinal torque on the optical fluence, (F), where the latter is proportional to the product of the intensity and pulse duration: (F \propto H_{peak}^{{2} \tau_{p}). Following the interaction, the accumulated torque results in a tilting of (\overrightarrow{M}). The longitudinal tilting is determined from the induced (\widehat{z}) component of (\overrightarrow{M}) after the interaction is completed, and is represented in normalized units, ({T}_{z}={M}_{z}/{M}_{s}). Figure 1b presents ({T}_{z}) as a function of (\eta) for different typical experimental conditions of (\alpha) and wavelength, (\lambda)1,7,13,39. (\eta) is varied by sweeping over the relevant range of ({H}_{peak}) values for each combination of (\alpha) and (\lambda). For clarity, we use the parameter ({\eta }_{H}) to indicate the sweeping over ({H}_{peak}). The figure readily shows that ({T}_{z}) is quadratic in ({\eta }_{H}), namely, ({T}_{z}) is linear in the optical intensity. Figure 1c illustrates the dependence of ({T}_{z}) on ({\tau }_{p}) for a constant ({\eta }_{H}) illustrating that ({T}_{z}) is linear in ({\tau }_{p}) in agreement with the trend reported experimentally in Refs.5,6. This behavior indicates that the optically induced torque builds up with each optical cycle. Since ({T}_{z}) scales with ({H}_{peak}}{2}) and ({\tau }_{p}), it also scales with the fluence (F). A linear dependence of the torque on (F) was also reported experimentally5,6,13,26,37 and was attributed to the non-linear susceptibility ({\chi }_{NL}^{{(2)})34,35,37. Following a detailed analytical derivation (see Supplemental Note 1), we find that (T_{z} = \frac{{\gamma}{2} }}{2\sqrt \pi }\frac{\alpha }{{f_{opt} }}H_{peak}^{{2} \tau_{p} \propto F\alpha /f_{opt}) (throughout the work we use ({\gamma }}{\prime}\approx \gamma)). This relation also shows that the torque is enhanced with (\alpha) and decreases with ({f}_{opt}). Interestingly, the AO-HDS was demonstrated in a variety of multi-layered material systems that consist of heavy metals such as Pt and Pd which possess large (\alpha)24,26,31,39. We point out that beyond the macrospin approximation, a spatial distribution could also affect the interaction. Such spatial dependence was recently investigated numerically by Zhang et al. in Ref.[50](https://www.nature.com/articles/s41598-025-24492-9#ref-CR50 “Zhang, Q., Lin, S., Zhang, W. Skyrmion generation through the chirality interplay of light and magnetism, arXiv preprint arXiv:2502.16197

(2025).“), where the optical profile and spin texture were calculated, enabling ultrafast excitation and control of the helicity of skyrmions using CP light.

The dependence ({T}_{z}\propto F) implies that the effect should be prominent for higher powers, where the pulse heating is higher, which may lead to electron heating due to absorption as explored by Kampfrath et al.20,21,22. Maehrlein et al.20 demonstrated that angular momentum transfer in yttrium iron garnet occurs in two stages that are characterized by distinct time constants: rapid spin-phonon energy equilibration within 1 picosecond, followed by angular momentum transfer to the lattice over 100 ns. Furthermore, Rouzegar et al.22 showed that ultrafast demagnetization and terahertz spin transport, previously considered distinct phenomena, share a common origin driven by a generalized spin voltage in a ferromagnet. Interestingly, in Ref.21 Chekhov et al. reported that the demagnetization does not depend on the wavelength and can equally take place with optical and terahertz (THz) excitations. In this case, the torque induced by the optical magnetic field may be described in more detail by the LLB equation43, where also longitudinal relaxation takes place. Such approach was explored by Korniienko et al.44, where the interaction with intense ultrashort THz pulses was studied in the framework of the LLB equation and a two temperature model coupling electrons and phonons. In Supplemental Notes 2 and 3 we include the anisotropy field and the longitudinal relaxation term, respectively, where it is seen that they have a negligible effect on the optically induced torque for the typical experimental settings we consider.

For a general polarization state, ({T}_{z}) is described by the difference between the RCP and LCP fluences. This is illustrated in Fig. 1d by plotting ({T}_{z}) as a function of the polarization state (\phi). It is readily seen that ({T}_{z}) vanishes for linearly polarized (LP) beams ((\phi =0^\circ ,180^\circ)) whereas for CP beams ((\phi =90^\circ ,270^\circ)) it is maximal, which is typical of AO-HDS1,6,26,51. For a general (\phi), ({T}_{z}\propto -\mathit{sin}\left(\phi \right)) (see Supplemental Note 1) which is proportional to ({I}_{RCP}-{I}_{LCP}) (see Supplemental Note 4). Hence, ({T}_{z}) is given by:

$$T_{z} = \frac{{\gamma^{2} }}{2\sqrt \pi c} \cdot \frac{\alpha }{{f_{opt} }}\left( {I_{RCP} - I_{LCP} } \right)\tau_{p} ,$$

(2)

where (c) is the speed of light. Equation (2) is valid for small angle dynamics corresponding to small values of (\eta) (see Supplemental Note 1).

The dependence of ({T}_{z}) on ({I}_{RCP}-{I}_{LCP}) also appears in Pershan’s phenomenological description of the IFE which was derived from the free energy of the crystal in the presence of the electrical component of the radiation, (\overrightarrow{E}). Accordingly, ({M}_{z}\propto {\chi }_{NL}^{{(2)}\left|\overrightarrow{E}\times {\overrightarrow{E}}}{*}\right|\propto {I}_{RCP}-{I}_{LCP})34,37. The potential function derived from Pershan’s Hamiltonian assumes a slowly varying optical intensity as compared to the thermal relaxation times of the system52. This assumption does not hold in the ultrashort timescales and several studies37,46,47,52,53 showed that the standard dependence on (\overrightarrow{E}\times {\overrightarrow{E}}^{*}) is incomplete in this limit. Reid46, Popova52, and Battiato et al.32 showed that on the subpicosecond timescales a stimulated magneto-Raman scattering process takes place which is known as the ultrafast-IFE.

The calculated transverse magnetization tilting also reproduces trends observed experimentally. It is represented by ({T}_{y}={M}_{y}/{M}_{S}) following the interaction. To illustrate this point, we examine the temporal evolution of (\overrightarrow{M}). Figure 2a presents the normalized (\overrightarrow{m}(t)=\overrightarrow{M}/{M}_{S}), for (\alpha =0.025), λ = 800 nm, ({\eta }_{H}=2\times {10}^{{-4}), and ({\tau }_{p}=540\text{ fsec}) resulting in ({T}_{y} \sim -2\times {10}}{-3}). Following the same numerical analysis and analytical derivation, we find that ({T}_{y}) is independent of (\alpha) such that ({T}_{y}\propto F/{f}_{opt}) (see Supplemental Note 1). The dependence (independence) of ({T}_{z}) (({T}_{y})) on (\alpha) was also observed experimentally by Choi et al.31 using time-domain vectorial torque measurements, where (\alpha) was varied by changing the metallic capping layer in ferromagnet (FM)/metallic bilayers.

Fig. 2

CP and LP single pulse dynamics. (a) Temporal evolution of (\overrightarrow{m}=\overrightarrow{M}/{M}_{s}) induced by an RCP Gaussian pulse under ({\eta }_{H}=2\times 1{0}^{-4}) and ({\tau }_{p}=540 \text{ fsec}, {t}_{peak}=2\text{ psec}). Top and middle panels depict the temporal evolution of the (x) and (y) components of (\overrightarrow{m}) and ({\overrightarrow{H}}_{opt}) in normalized units, and the bottom panel depicts ({m}_{z}). Inset: zoomed in dynamics of ({m}_{z}) following the pulse. (b) Dynamics under LP pulse. (\alpha =0.025) and λ = 800 nm in (a) and (b).

Interestingly, in Ref.30, Ali et al. demonstrated that an effective IFE magnetic field can be induced even under an LP beam. The effect arises from the transfer of orbital angular momentum from a beam with a helical wavefront, where each of the orthogonal Laguerre-Gaussian modes composing the helical wavefront transfers its well-defined photon orbital angular momentum to the plasma. This result shows that angular momentum could also be transferred by means other than the circular polarization of light and stimulates the investigation of LP beams in our case. For comparison, in Fig. 2b we examine the temporal evolution of (\overrightarrow{M}) driven by a single LP ({\overrightarrow{H}}_{opt}) pulse ((\phi =0^\circ)), under the same conditions used in Fig. 2a. It is readily seen that although at the end of the interaction the net induced torque is zero, (\overrightarrow{M}) undergoes a non-trivial dynamical evolution. Further investigation of the dependence of the LP case on the pulse power, duration, and polarization direction is presented in Supplemental Note 5, illustrating that the polarization direction affects the dynamical evolution while the resultant torque remains zero.

The torque induced by a single pulse can be equivalently achieved by applying multiple pulses whose total fluence equals that of the original pulse. Figure 3a presents the temporal response of (\overrightarrow{m}) to (10) identical ({\overrightarrow{H}}_{opt}) pulses, applied as in Fig. 2a, except that each pulse has one tenth of the duration ({\tau }_{p}), and an arbitrary carrier phase. It is seen that following the entire interaction, the accumulated torque is equal to the torque induced by the original single pulse of Fig. 2a and is independent of the relative carrier phases. Figure 3b presents ({T}_{z}) induced by multiple RCP ({\overrightarrow{H}}_{opt}) pulses as a function of ({\eta }_{H}) and the number of applied pulses. It is seen that ({T}_{z}) is linear in the number of pulses and quadratic in ({\eta }_{H}). In this general case, ({T}_{z}=\frac{1}{2\sqrt{\pi }\alpha }\frac{#pulses\times {\tau }_{p}}{{t}_{cycle}}{\eta }^{2}) (see Supplemental Note 6) such that the optically-induced torque builds up with each applied pulse as also reported experimentally5,6,51. The total torque can be induced either by a single pulse or by multiple pulses which further demonstrates that ({\rm T}_{z}) scales with the accumulated exposure time.

Fig. 3

Multi-pulse and CW regimes. (a) Temporal evolution of (\overrightarrow{m}) under ({\eta }_{H}=2\times 1{0}^{{-4}) and ({\tau }_{p}=54 \text{ fsec}), induced by 10 RCP Gaussian magnetic pulses. Top and middle panels depict the temporal evolution of the (x) and (y) components of (\overrightarrow{m}) and ({\overrightarrow{H}}_{opt}) in normalized units, and the bottom panel depicts ({m}_{z}). For visibility, brown dashed lines representing ({T}_{z}) induced by each pulse are added and the interaction with the first pulse is highlighted. (b) Normalized (|{T}_{z}|) after the application of RCP pulses as a function of ({\eta }_{H}) and the number of pulses, where ({\tau }_{p}=54 \text{ fsec}) for each pulse. Red and blue curves correspond to quadratic and linear fits, respectively, and are guides to the eye. (c) Temporal evolution of (\overrightarrow{m}) induced by a CW RCP magnetic field under (\eta ={10}}{-7}). Top, middle, and bottom panels depict the evolution of the (x), (y), and (z) components of (\overrightarrow{m}), respectively. In (a) to (c), (\overrightarrow{m}) is initialized in (\widehat{x}), (\alpha =0.025), and λ = 800 nm as in Fig. 2.

The dependence on the exposure time suggests that the effect may be also relevant for longer pulses reaching the CW limit as reported recently by Stenning et al. 7. The dynamics induced by a rectangular quasi-CW pulse are depicted schematically in Fig. 3c by introducing an RCP CW beam, ({\overrightarrow{H}}_{opt CW}\left(t\right)), at 800 nm for a duration of ({t}_{CW\