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Abstract
We theoretically investigate a resonant inverse Faraday effect within individual coaxial nanoapertures. Upon illumination with circularly polarized light, resonant gold coaxes are shown to develop an optomagnetic field that is controllable by the helicity of the light. This magnetic field is found to reach 0.13 T upon excitation at an intensity of 0.5 · 1011W.cm−2 that is typical from sub-ps light pulses. At an intensity of 2.4 · 108W.cm−2 (consistent with the CW regime), we obtain a static magnetic field of about 1 mT, leading to a helicity-dependent magnetic force of 4.5 · 106 N onto a point-like magnetic dipole of unit moment. Given their submicron footprint, individual coaxial nanoapertures open new prospects towards ultrafast and polarization-controlled tunable magnetism on the nanoscale, thus potentially impacting a large panel of application and techniques including all optical magnetization switching, spin-wave excitation and optomagnetic tweezing of nano-objects.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Magnetic fields are at the heart of a large variety of techniques and applications, including spintronics, data storage, magnonics and medical imaging. The interplay between light and magnetism in matter provides ways to control light with magnetic field and, in turn, to trigger magnetic effects with light [1].
As an example, a static magnetic field can be generated in matter under illumination with a circularly polarized light. The underlying physical phenomenon, known as the inverse Faraday effect, has attracted much attention for its unique ability to attain ultrafast all-optical magnetization control in ferromagnets, thus opening the prospect of a magnetic data storage with unprecedented speed [2–4] and a non-contact excitation of spin-waves [5–9].
The inverse Faraday effect is a nonlinear optical phenomenon leading to a magnetism proportional to the optical intensity [10,11]. Plasmonic nanostructures have been recently investigated to enhance the inverse Faraday effect in non-magnetic metals [12–21] and in hybrid structures combining non-magnetic metals and magnetic materials [22–27]. Plasmonic nano-antennas made of a single metal nanostructure or closely spaced nanoparticles hold promises for enhancing and confining down to sub-micron scale optomagnetic effects and fields, thereby responding to the requirement of an ultrafast and highly localized magnetism with a minimum energy consumption.
Here, we investigate a resonant inverse Faraday effect in individual gold coaxial aperture nanoantennas. A coaxial nanoantenna is an axis-symmetrical nanoaperture which develops optical resonances on the basis of the excitation of surface plasmons that are tightly confined in between its two face-to-face cylindrical edges [28–31]. On the basis of a semi-classical hydrodynamic model of the free electron gas of metal [19], we study the generation of an optically-induced magnetic field within plasmonic coaxial nanoapertures upon illumination with a circularly polarized light. This optomagnetic field can be reversed by flipping the helicity of the incoming light. Simplifying the quantum hydrodynamic model required to rigorously describe optomagnetism in noble metals [14,17,32] helps leaving basic nanoparticle geometries and addressing optomagnetism in more complex 3D nanostructures usually obtained from top-down nanofabrication techniques. The design of optimized structures for all-optical magnetization switching, spin-wave excitation and nanoparticle tweezing may thus become possible.
2. Simulations and design
All numerical simulations are carried out using commercial 3D finite difference time domain method (FDTD) from Synopsis. The coaxial nanoapertures are considered to be engraved in a thin gold film lying on a semi-infinite glass substrate of refractive index equal to 1.5 (Fig. 1). The radii of the two cylindrical edges defining the nanostructures are labeled $r_1$, $r_2$ from the center to the side ($r_1<r_2$). The nonuniform grid resolution within the computation volume varies from 10 nm for portions at the periphery of the simulation to 0.2 nm within the nanoaperture. Four coax geometries are investigated, three of them are defined with a film thickness of 50 nm and three gap widths $r_2$-$r_1$ of 10, 20 and 50 nm. The last geometry refers to a 20-nm thick gold film and a gap width $r_2$-$r_1$ of 10 nm.
Fig. 1. Schematics of the configuration under study: (a) top view of a coaxial nanoaperture in a thin gold film lying on a glass substrate. (b) the nanostructure is excited from the substrate with a circularly polarized focused beam ($\lambda$=800 nm, spot size of one $\lambda$). The nanostructure is suggested to optically induce an optomagnetic field by virtue of the inverse Faraday effect.
As a first numerical step, we design all the coaxial nanoapertures to be resonant at $\lambda$=800 nm where $\lambda$ is the vacuum wavelength (Fig. 2). For each nanoaperture geometry, the optical spectrum is obtained from a Gaussian excitation described by a short single temporal pulse. The time-varying electric field is calculated at a single cell located 0.4 nm away from the inner edge, in the metal. The spectral response is then calculated by Fourier transforming these results. The spectrum is finally normalized with that obtained in the same numerical conditions but without metal. The four nanoaperture geometries are detailed in the figure inset. For gap widths $r_2$-$r_1$ equal to 10, 20 and 50 nm, the coaxial nanoapertures in a 50-nm thick gold film show an outer radius $r_2$ of 45, 65 and 115 nm, respectively. Such aperture geometries are experimentally realistic with top-down nanofabrication facilities [30,31]. The light field enhancement observed within the metal originates from the optical capacitive effect bound to the air gap in between the two closely spaced cylindrical edges [28]. This plasmonic phenomenon strengthen the optical electromagnetic field and charge accumulation at the two face-to-face edges, which is in favor of optical nonlinearities such as inverse Faraday effect.
Fig. 2. Surface plasmon resonance spectra of four coaxial nanoapertures. Three of them are engraved in a 50-nm thick gold film and show gap widths $r_2$-$r_1$ equal to 10 nm (blue curve), 20 nm (red curve) and 50 nm (green curve). The fourth nanoaperture is defined by a 10-nm thick gold layer and gap size of 20 nm (black curve).
In a second time, we simulate the linear optical response of the nanoapertures upon excitation with a monochromatic wave at $\lambda$= 800 nm. Each nanoaperture is illuminated with a left-handed circularly polarized Gaussian beam (beam waist equal to $\lambda$) propagating along the symmetry axis of the nanostructure. The beam waist is projected onto the backside of the nanostructure, in glass. According to our convention, left circular polarization corresponds to an electromagnetic field rotating counterclockwise when ones look toward the propagation direction of the incoming light beam. We calculate the optical electric and magnetic field distributions across the nanostructures in the linear regime.
Then, the nonlinear response of the coaxial nanoapertures is predicted by applying the hydrodynamic model introduced in Ref. [19] to the FDTD-calculated optical fields. A more detailed description of our formalism in given in the Appendix. The distribution of optically-induced drift current density across the nanostructure is deduced from the bulk and surface contributions of the source terms given in Eqs. (8) and 9. The conductivity $\gamma _\omega$ of the metal at optical frequencies is written as:
where $\varepsilon _\omega = \varepsilon ' + i~\varepsilon ^{''}$ is the complex relative permittivity of the metal ($\varepsilon _0$ is the permittivity of vacuum). $\varepsilon _\omega$ is an experimentally accessible optical parameter of materials which merges conductivity from free electrons and absorption from bound electrons (e.g., via electronic inter-band transitions). Deducing $\gamma _\omega$ from experimental $\varepsilon _\omega$ thus requires to work in the near-infrared spectral domain where the free electrons mainly drive the optical properties of the metal (i.e., $\varepsilon _\omega$ is accurately described with Drude model) [33,34]. This condition is fulfilled in our study.Given the cylindrical symmetry of our system, the optomagnetic field is supposed to mainly arise from the azimuthal component of the nonlinear source terms. The contribution of the longitudinal and radial components of the nonlinear source terms (along (0z) and (0r), respectively) are thus neglected.
Finally, the optically-induced magnetic field is calculated from the current density using the well-known Biot and Savart law:
The plasmon-induced magnetic moment $\boldsymbol {\mu }$ of the nanoaperture can be deduced from the drift current density $\mathbf {j_d}$ [35]:
The magnetic force exerted on a particle of intrinsic magnetic moment $\mathbf {m}$ reads $\mathbf {F}=\nabla \left (\mathbf {m} \cdot \mathbf {B} \right )$.
3. Results and discussions
In our hydrodynamic model, a monochromatic light is considered, i.e., either optical waves propagating in the CW regime or sufficiently long laser pulses (of sufficiently narrow spectra) so that all frequency dependence can be neglected.
3.1 Optomagnetism in pulse regime
Optically-induced magnetic effects are usually studied in the sub-ps pulse regime to meet the requirements of high light intensity and ultrafast magnetic response. Pulses showing peak intensity of 0.5 10$^{11}$W.cm$^{-2}$ can reasonably be launched onto a plasmonic gold nanoantenna to induce optical nonlinearities without damaging the nanostructure [22,36,37].
As an example, such an intensity level would be reached in a 50-fs laser pulse at a fluence of 3.1 mJ.cm$^{-2}$. Using the FDTD method, we verified that the distortion at a 50-fs pulse by our coaxial nanoapertures can be neglected. In pulse regime, the inverse Faraday effect leads to a transient magnetic field on a typical time duration defined by the pulse envelope. The underlying optical rectification process [38] has been investigated in thin metal films [39–42].
The optomagnetism is characterized in a (x0z)-plane that includes the axis of symmetry of the coaxial nanoaperture. Figure 3 represents the amplitude of the electric optical field and the total optically-induced drift current across three individual coaxial nanoapertures illuminated with a circularly polarized light of intensity 0.5 10$^{11}$W.cm$^{-2}$ at $\lambda$=800 nm. drift current values are obtained by calculating $\mathbf {j^b_d}+\left [\mathbf {j^s_d}\right ]_\theta$ over 0.1$\times$0.1 nm$^2$ areas along the cross-section of the nanostructures. Positive/negative values in the figures refer to drift currents that are parallel/anti-parallel to the (0y)-axis (as defined in Fig. 3(e)). Calculations are realized with a conductivity of gold at the zero frequency of $\gamma _0$=4.11 10$^7$S.m$^{-1}$ [43,44], an effective mass $m_e$ of the electron of $0.99\times 9.1091$ 10$^{-31}$ kg [33,45] and a density $n_0$ of free electrons of 5.9 10$^{28}$ m$^{-3}$ [46]. We see in Figs. 3(a), (c) and (e) that the plasmon-induced inverse Faraday effect induces mainly drift surface currents (as already shown in Refs. [16,19,32]) which are maximum at metal corners where surface plasmons are locally confined and enhanced (see Figs. 3(b), (d) and (f)) [19]. The maximum drift current reaches 25, 10 and 3 $\mu$A (across a 0.1$\times$0.1 nm$^2$ area) for the 10, 20 and 50-nm gap coaxes, respectively. The drift current decay observed as the aperture gap broadens (Figs. 3(a), (c) and (e)) results from a drop of the nanoaperture resonance (see Fig. 2) accompanied by a decrease of the electric optical field right at the nanoaperture gap, in the metal (Figs. 3(b), (d) and (f)). This is the manifestation of a weaker capacitive effect in between the two facing cylindrical edges of the nanostructure, resulting in a decrease of both charge accumulation and ponderomotive forces at metal interfaces. Note that the amplitude and handedness of the drift current density within the nanoapertures can be tuned by controlling the helicity of the incoming light.
Fig. 3. (a), (c) and (e) Distribution in the (x0z)-plane of the plasmon-induced drift current distribution within the 10, 20 and 50-nm gap coaxes, respectively. Each current value is calculated across a 0.1$\times$0.1 nm$^2$ area. Positive and negative values refer to drift currents that are parallel and anti-parallel to the (0y)-axis, respectively (see coordinate frame in (e)). (b), (d) and (f) Amplitude of the electric optical field in the same (x0z)-plane for the 10, 20 and 50-nm gap coaxes, respectively. Illumination is realized with a circularly polarized light of intensity 0.5 10$^{11}$W.cm$^{-2}$ at $\lambda$=800nm. The sense of rotation of the drift current can be changed by flipping the polarization handedness of the incoming light. Scale-bars: 20 nm.
Plasmon-induced optomagnetic fields (Fig. 4) are maximum at the two metal corners in contact to the glass substrate, where they undergo steep gradients. They reach 0.13 T for the 10-nm gap coaxial nanoaperture (Figs. 4(a) and (b)) and decreases down to 0.065 T and 0.03 T, when the gap is enlarged to 20 nm (Figs. 4(c) and (d)) and 50 nm (Figs. 4(e) and (f)), respectively. From Eq. (3), we predict a corresponding optically-induced magnetic moment oriented along (0z) and of maximum amplitude of the order of 1.3 10$^{-20}$, 9.7 10$^{-21}$ and 5.84 10$^{-21}$ J.T$^{-1}$ for the 10, 20 and 50-nm gap coaxial nanoapertures, respectively. This optomagnetic effect can be reversed simply by flipping the helicity of the incoming light.
Fig. 4. Distributions of the optomagnetic field in the longitudinal cross-section (x0z) of the coaxial nanoapertures of gap size equal to (a,b) 10 nm, (c,d) 20 nm and (e,f) 50 nm. The amplitude of the optomagnetic field is plotted (a,c,e) in linear scale and (b,d,f) in logarithmic scale. In (b,d,f) are superimposed the local optomagnetic field orientation represented with black arrows. Illumination is realized with a circularly polarized light of intensity 0.5 10$^{11}$W.cm$^{-2}$ at $\lambda$=800nm. The optomagnetic field can be reversed by flipping the helicity of the incoming light. Scale-bars: 20 nm.
The optomagnetic field is mainly concentrated on the backside of the nanostructure (in contact to the substrate), which is an advantage for applications requiring magnetic effects in the substrate, such as spin-wave excitation or all-optical magnetization switching.
In order to enhance optomagnetic fields on the front side, one can consider a thinner gold film. Figure 5 represents the plasmon-induced optomagnetic field across a 10-nm gap and 20-nm high coaxial nanoaperture. The structure is resonant at $\lambda$= 808 nm ($r_1$=60nm and $r_2$=80nm). The optomagnetic field shows a maximum value of about 0.1 T on its backside that is reduced by a factor of 3.1 on its front side, instead of a factor of 5.8 in the case of the 10-nm gap and 50-nm high coax.
Fig. 5. Distribution of the optomagnetic field in a longitudinal (x0z) cross-section of a coaxial nanoaperture defined by a gap size of 10 nm and gold film thickness of 20 nm. (a) amplitude in linear scale. (b) Amplitude in logarithmic scale and local optomagnetic field orientation represented with black arrows. Illumination is realized with a circularly polarized light of intensity 0.5 10$^{11}$W.cm$^{-2}$ at $\lambda$=800nm. The optomagnetic field can be reversed by flipping the helicity of the incoming light. Scale-bars: 10 nm.
3.2 Optomagnetism in CW regime
In a CW excitation regime, an incoming focused light beam of 1 Watt induces an on-axis intensity at focus of 2.4 10$^{8}$W.cm$^{-2}$ for a spot size of one wavelength. At such an intensity, a maximum optomagnetic field reaching 0.94 mT is predicted in our 10-nm gap and 20-nm high coaxial nanoaperture. Figure 6(a) and (b) report in the (x0z)-plane the resulting magnetic force exerted onto magnetic dipoles oriented along (0x) and (0z), respectively. The two magnetic dipoles are of unit intrinsic moment (1 J.T$^{-1}$). The amplitude of the magnetic force reaches 9.8 10$^{5}$ N and 3.2 10$^{6}$ N at the front and back sides of the nanoapertures, respectively. The local maxima of the force are located at the metal corners. A nanoscale magnetic tweezing could thus be possible in plasmonic nanostructures. The optomagnetic field would then provide an additional polarization-controlled tunable force to the ones involved in pure optical tweezing by the nanoaperture itself [30,31], thereby offering new degrees of freedom in nano-object trapping and manipulation.
Fig. 6. Optically-induced magnetic force plotted in the (x0z)-plane of a coaxial nanoaperture defined by gap width and height of 10 nm and 20 nm, respectively. Amplitude of the optomagnetic force is represented in logarithmic scale while its local orientation is shown with white arrows. The force exerts on a magnetic dipole of unit moment $\mathbf {m}$ that is oriented (a) along (0z) and (b) along (0x). The magnetic dipole is represented in (a) and (b) with a bigger white arrow and the letter "m". The amplitude and local orientation of the optomagnetic forces car be controlled with the helicity of the incident light. Scale-bars: 10 nm.
4. Conclusion
On the basis of a simplified hydrodynamic model of the free electron gas of a metal, we investigate the generation of an optomagnetic field (generated from the inverse Faraday effect) in plasmonic coaxial nanoapertures upon illumination with a circularly polarized light. We predict a optomagnetic field of 0.13 T at laser fluence below nanoaperture damage threshold. We also show that the substrate introduces an important asymmetry of the optomagnetic response of the plasmonic nanostructure. For a 50-nm thick gold film, the optomagnetism is mainly localized within the substrate, which appears to be advantageous for many applications. Note that fabricating metal nanostructures with sharp corners is challenging. Therefore, one should expect lower optomagnetic fields in fabricated coaxial nanoapertures. However, since very sharp metal corners are achievable in contact to a substrate [47], a minimum discrepancy between experimental results and theoretical predictions could be expected. Reducing the gold film thickness to 20 nm is found to better homogenize the magnetic effects along the nanoaperture. Optomagnetism in plasmonic nanoapertures may impact a broad field of applications and techniques including spintronics, magnonics and data storage via the development of on-chip nanoscale plasmonic-magnetic architectures. Optomagnetism may also provide new degrees of freedom in nano-object tweezing based on the combination of optomagnetic and pure optical forces. Interesting analogy with the plasmonic Aharonov-Bohm effect may also be found [48]
Appendix: Simplified hydrodynamic model to predict optomagnetism in metals
In a hydrodynamic approach, the free electron dynamics can be described from the Euler’s equation:
In the time harmonic (i.e., monochromatic) regime, the current density reads:
where $\omega$ is the angular frequency, $t$ is time, $c.c$ is the complex conjugate. $\mathbf {j}_{\boldsymbol {\omega }}$ and $\mathbf {j_{NL}}$ are the linear and nonlinear contributions of the current density, respectively. $\mathbf {j_{NL}}$ is considered to be small as compared to $\mathbf {j}_{\boldsymbol {\omega }}$. In the following, we focus on the nonlinear optical process defined by the drift current density $\mathbf {j_d}=\langle \mathbf {j_{NL}}\rangle$ where the operator $\langle \rangle$ denotes time averaging. $\mathbf {j_d}$ originates from an optical rectification process [38].In a perturbative approach, Eq. (4) expressed in terms of current density leads to the following set of equations:
The right side of Eq. (7) combines four nonlinear optical terms. We have the magnetic Lorentz force term $\mathbf {j}_{\boldsymbol {\omega }} \times \mathbf {H}^*_{\boldsymbol {\omega }}$, the nonlinear quadrupolar or Coulomb contribution $\left (\mathbf {\nabla }\cdot \mathbf {j}_{\boldsymbol {\omega }} \right ) \mathbf {E}^*_{\boldsymbol {\omega }}$ and the convective term $\left ( \mathbf {j}_{\boldsymbol {\omega }} \cdot \mathbf {\nabla }\right )\mathbf {j}^*_{\boldsymbol {\omega }} + (\mathbf {\nabla } \cdot \mathbf {j}_{\boldsymbol {\omega }})\mathbf {j}^*_{\boldsymbol {\omega }}$. The electron gas pressure leads to the first harmonic nonlocal term proportional to $\beta ^2\mathbf {\nabla }(\mathbf {\nabla } \cdot \mathbf {j}_{\boldsymbol {\omega }})$ (see Eq. (6)).
Using Eqs. (6) and 7 in their present form, the nonlinear problem remains complex to solve. To simplify the equations, a bulk-surface model of the metal can be considered, as suggested in the description of second and third harmonic generation [49,50]. Following this approach, two complementary regions within the metal are defined: a "surface layer" right beneath the metal boundaries whose thickness matches the Thomas-Fermi screening length ($\lambda _{TF}\simeq$ 0.1 nm for gold) and the rest of the metal, defined as the "bulk".
Drift current density within the metal bulk
As discussed by Sipe et al. [49], it is possible to neglect the pressure terms ($\beta \rightarrow 0$) in the metal bulk as it leads to corrections of the order of $(\lambda _{TF}/\lambda )^2\ll 1$ in both linear and nonlinear optical regimes. The linear contribution of the current density then takes the simple macroscopic form corresponding to the local response limit: $\mathbf {j}_{\boldsymbol {\omega }}= \gamma _{\omega } \mathbf {E}_{\boldsymbol {\omega }}$ where $\gamma _\omega = \gamma _0/(1-i \omega \tau )$ and $\gamma _0 = n_0~e_e^2~\tau /m_e$ are the dynamic and static conductivities of the metal, respectively. Then the bulk contribution of the drift current density reads:
Drift current density at metal interfaces
Within the surface layer, the pressure terms cannot be neglected as electron-electron interactions are strongly enhanced [49–52]. Moreover, in case of a neglected electron pressure at interfaces, the theory would become inherently ambiguous as surface sources would be defined by Dirac delta functions [49].
To solve our optical rectification problem in the surface layer, the nonlocality of the hydrodynamic model (i.e., the spatial derivatives of the fields) requires an additional boundary condition to those already used together with Maxwell’s equations. Within the surface layer, the linear current density $\mathbf {j}_{\boldsymbol {\omega }}$ is considered to have a component $\boldsymbol {j_{\omega }^T}= \boldsymbol {N} \times \mathbf {j}_{\boldsymbol {\omega }}$ ($\boldsymbol {N}$ is the surface normal) along the interface whereas its normal component $j_{\omega }^N$ is supposed to decay to zero [50]. A vanishing $j_{\omega }^N$ at the vicinity of the surface is deduced from the continuity equation and Gauss’s theorem and refers to a neglected "spill-out" of the electron density at the interface [51,52].
Defining $\xi$ as the spatial coordinate normal to surfaces so that the metal bulk is located at $\xi <0$ and the surface layer corresponds to $0<\xi <\lambda _{TF}$, we have $\boldsymbol {j_{\omega }^T}(\xi ) \approx \boldsymbol {j_{\omega }^T}(0^-)$ and $j_{\omega }^N=j_{\omega }^N(0^-) \sigma (\xi )$, where $\sigma$ is a decaying function defined by $\int _0^{\lambda _{TF}} \sigma '(\xi )d\xi =-1$ where $\sigma '$ is the derivative with respect to $\xi$ [50].
In the cylindrical coordinates $(r,\theta ,z)$, the azimuthal component of the surface contribution of the drift current density takes the form [19]:
Equation (9) implies that the spatial derivatives of the fields along the surface are negligible. Terms $i \gamma _0/(\omega \tau \gamma ^\ast )$ and $1$ in Eq. (9) refer to the nonlinear quadrupolar and convective source terms, respectively.
Funding
Conseil régional de "Bourgogne Franche-Comté"; EIPHI Graduate School (ANR-17-EURE-0002); Agence National de le Recherche (ANR-18-CE42-0016).
Acknowledgments
The authors are indebted to François Courvoisier and Ulrich Fischer for helpful discussions.
Disclosures
The authors declare no conflicts of interest.
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