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Abstract
A significant enhancement of the longitudinal magneto-optical effect is demonstrated numerically and experimentally in transmission, and for small angles of incidence, through a subwavelength resonant structure consisting of a dielectric grating on top of a magneto-optical waveguide. The enhanced polarization rotation is associated with a high transmittance. These low footprint devices may thus be suitable for applications like magnetic field sensors or in non-destructive testing.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Magneto-optical (MO) effects [1] are described by a rotation of the plane of polarization (Faraday, polar or longitudinal Kerr) or by an intensity modification (transverse Kerr) of the reflected or transmitted light through a magnetized MO material. These effects are used for applications such as optical isolators [2,3], which are essential elements in laser systems. They can be used also in magnetic field sensors [4,5], MO data storage [1] and biosensors [6].
Currently, having an integrated device with high MO effects attracts much attention. But the problem is that the MO effects are weak for small volumes and small angles of incidence. Hence, the enhancement of these effects is required.
The enhancement of MO response can be reached by subwavelength magneto-plasmonic structures: combination of metallic grating with MO dielectric material. The excitation of eigenmodes of the structure, such as propagated surface plasmon polaritons (SPP) [7], localized surface plasmon (LSP) [8], cavity modes [9] or waveguide modes [10], is responsible of the resonant enhancement of the MO response in these structures. One of the simplest magneto-plasmonic structures is consisting of a gold grating deposited on a layer of BIG. Enhancement of MO Faraday effect [7,8,10], MO Kerr effects [11,12] were demonstrated by this type of structure. Moreover, enhancement of MO effects can be reached theoretically or experimentally by more complex magneto-plasmonic structures as demonstrated in the works of Floess et al. [13], Lei et al. [14,15] and Li et al. [16].
Another way to exalt the MO effects is by using a subwavelength dielectric resonant guided-mode grating: dielectric MO waveguide surrounded by a dielectric grating. These structures are based on the phenomenon of the waveguide mode resonance [17], which results in a dip in the transmittance spectrum [18]. The dielectric grating plays a twofold role. On the one hand, it couples the light into the MO waveguide with a certain coupling condition given by:
where $\beta$ is the propagation constant of the waveguide mode, $n_{1}$ is the refractive index of the incidence medium, $\theta _{inc}$ is the incident angle, $m$ is the diffraction order, $\Lambda$ is the period of the dielectric grating and $\lambda _{0}$ is the vacuum wavelength for the incident light.Due to the trapped light in the MO waveguide, the propagation length is increased as compared to a single pass through the MO film. Hence, the interaction between the light and the MO material is enhanced in the resonant structures resulting in higher MO rotations, in comparison to a MO film without grating.
On the other hand, the periodic nanostructuring with optimized opto-geometric parameters engenders the phase matching condition ($\beta _{TE}=\beta _{TM}$). It is well known that the TE-TM mode conversion, in other word the Faraday, polar and longitudinal MO Kerr effects, is maximum when the phase matching condition is satisfied [19].
An enhancement of Faraday and polar Kerr effects have been theoretically demonstrated by Bai et al. [20] through a 2D subwavelength dielectric grating with optimized parameters. The use of symmetric 2D grating allows to fully degenerate the polarization at normal incidence: in such a case TE and TM resonances will always match. Another theoretical enhancement of the MO effects (Faraday and three Kerr configurations) was obtained by Gamet el al. [21]. The work was done by a 1D subwavelength dielectric structure and the parameters were optimized to full-fill the phase matching condition, $\Delta \beta =0$. Maksymov et al. [22], have worked with the transverse Kerr [1], and they demonstrated by numerical simulations a variation of $4\%$ of the reflectivity through a dielectric structure consisting of alternating MO material (Bi:YIG) and nonmagnetic dielectric (Si) nanostripes. An enhancement of two orders of magnitude of the MO effect was achieved in comparison to the continuous film. Here, the condition of phase matching is not required since the transverse Kerr is only for the TM polarization [1].
Other possible structures to enhance the MO effects are magnetophotonic crystals [23–25] or dielectric MO metasurfaces [26].
In this paper, numerical and experimental significant MO responses in the longitudinal configuration [1] and in transmission, are presented for small angles of incidence (AOI) by a simple 1D dielectric resonant guided-mode grating.
2. Materials and methods
The structure geometry is depicted in Fig. 1(a). It consists of a photoresist (PR) grating deposited on a thin layer of a MO composite material. The used PR is the S1805. It is a positive resin, which has been chosen for several reasons: first, this resin has a good adhesion to the MO film without the requirement of an adhesion promoter. In addition to that, it is possible to dilute the S1805 in ethyl-lactate in order to change its viscosity and thus modify the thickness of the deposited layer. This important feature allows us to control the grating height and width. Moreover, the PR requires a low annealing ($60^{\circ }$C) compared to the other materials like titanium oxide or silicon nitride. Finally, the PR gratings are simple to fabricate which leads to low costs devices.
Fig. 1. (a) Schematic drawing of the sample geometry, the incidence conditions and the longitudinal magneto-optical effect in transmission. The geometrical parameters of this structure are: $t_{MO}=460nm$, $t_{PR}=130nm$, $h_{PR}=300nm$, $w=400nm$ and $\Lambda =1000nm$. (b) 3D image of the photoresist grating’s topography obtained by AFM measurements.
The structure was fabricated as follows: first a layer of MO composite with thickness $t_{MO}$ was deposited on a glass substrate by dip coating then treated thermally at $90^{\circ }$C for 1 hour. After that, the PR grating with height $h_{PR}$, width $w$ and period $\Lambda$, was structured on the top of the MO film by a classical photolithographic process using a periodic quartz-supported chromium amplitude mask. In order to have the two guided modes (TE and TM) in the structure, the grating is not dug to the bottom and a thin layer of PR with thickness $t_{PR}$ is kept since its refractive index is close to that of the MO film.
The MO composite was prepared by a liquid sol-gel preparation of Tetraethyl orthosilicate ($C_{8}H_{20}O_{4}Si$), doped by a ferrofluid containing cobalt ferrite ($CoFe_{2}O_{4}$) nanoparticles (NPs). For details on the MO composite elaboration, the readers are referred to [25].
With this MO sol-gel, it is possible to deposit uniform thin layers on a large scale substrate and with a $100^\circ C$ thermal treatment. Such composite has been used for the fabrication of integrated MO converters [27] and 3D magneto-photonic crystals [28]. Also, it can be employed to impregnate 1D, 2D [29] and 3D micro/nano-structured templates [28].
To study the behavior of the refractive index and the thickness of the MO thin layers as a function of the concentration ($\phi$) of NPs, this latter was varied from $8\%$ to $26\%$ by modifying the volume fraction of the ferrofluid. For the resonant structure, under consideration further in the text, the concentration was chosen to be the highest one: $\phi =26\%$. This volume fraction of NPs $\phi$ in the final composite, is deduced from specific Faraday rotation measurement.
The refractive indices were measured by ellipsometry (Horiba Jobin Yvon UVISEL) for a large wavelength range (280-1700nm). At the wavelength $\lambda =1550nm$, $n_{PR}=1.59$ and $n_{Substrate}=1.51$. The refractive indices of the MO films ($n_{MO}$) for different concentrations of NPs in the composite are plotted in Fig. 2(a) at $\lambda =1550nm$. As seen in this figure, $n_{MO}$ is dependent on $\phi$ with a linear variation. The index evolves from a value of $1.48$ for $\phi =8\%$ to a value of $1.61$ for $\phi =26\%$. The extinction coefficient of the MO film ($k_{MO}$) is also proportional to the NPs concentration and for $\phi =26\%$, $k_{MO}=0.0044$ at $\lambda =1550nm$. Hence, an important feature of this MO composite material is that the complex refractive index can be tuned. Therefore, the diagonal elements of the MO film’s permittivity ($\epsilon _{11}$, $\epsilon _{22}$, $\epsilon _{33}$) evolve with $\phi$ and for the chosen concentration of $26\%$ in this work, $\epsilon _{11}=\epsilon _{22}=\epsilon _{33}=2.5921 - i0.0142$. The MO layer’s thicknesses ($t_{MO}$) were measured also by ellipsometry. For the chosen $\phi =26\%$, $t_{MO}=460nm$. The values of the MO refractive index and the MO film thickness were chosen to ensure the confinement of the light in the MO waveguide. The off-diagonal elements of the tensor permittivity of our MO material for $\phi =26\%$ are: $\epsilon _{23}=-\epsilon _{32}=-i0.0064$ at $\lambda =1550nm$ [21]. Out of the grating area, the thickness of the non-insolated photoresist (attributed to $t_{PR}+h_{PR}$) was measured also by ellipsometry and it is equal to $430nm$. To obtain a resonance around $\lambda =1550nm$ with the given refractive indices and small AOI, the period was chosen to be $1000nm$. Hence, the structure is a subwavelength grating: only the zero diffraction order exists in the substrate and the upper air cover, and the $\pm 1$ diffracted orders in the MO waveguide. The grating’s parameters were measured by Atomic Force Microscopy (AFM) as illustrated on Fig. 1(b) and the period was confirmed by Littrow mount measurements: $\Lambda =1000nm$, $h_{PR}=300nm$, $w=400nm$. The values of the grating height and width were chosen to ensure the coupling of the incident light into the MO waveguide.
Fig. 2. (a) Measurements of the refractive index at $\lambda =1550nm$ with a linear trend line for different concentrations of nanoparticles. (b) Example of measurements of the polarization rotation at a fixed wavelength ($\lambda =1580nm$) for a varied applied magnetic field, in the longitudinal configuration. Inset: the hysteresis loop for normalized Faraday rotation.
The polarization rotation was measured at room temperature with a homemade MO setup described as follows: the light coming from a tunable laser ($1480-1630nm$) is linearly polarized after passing through a polarizer. The plane of this linear polarization is rotated after passing through the magnetized resonant structure. This latter is fixed on a support with five axes insuring a wide swipe on its surface and a precise angle of incidence and it is placed between two pieces of an electromagnet generating a longitudinal magnetic field $\overrightarrow {B}$ (parallel to the incident and sample planes as seen in Fig. 1(a)). After that, the light passes through a photoelastic modulator, with a modulation frequency F (50 kHz), and an analyzer before being finally detected by the photodetector. The detected signal is analyzed by a synchronous detection and after an automatic calibration the polarization rotation is calculated with a precision in order of a thousandth of a degree. For more information concerning the MO characterizations, the readers are referred to [28].
Due to the geometry of the electromagnet in the longitudinal configuration, $\overrightarrow {B}$ can be varied only in the range of [−360mT ; +360mT] which is sufficient to reach 80% of the polarization rotation’s saturation. This is verified by the inset of Fig. 2(b), where an hysteresis loop of normalized Faraday rotation [1] is plotted. Here, due to the geometry of the electromagnet in the Faraday configuration setup, the magnetic field can reach a maximum value of 800mT. As seen in this figure, for $B=360mT$, 80% of the saturated polarization is achieved.
For every wavelength, the longitudinal applied magnetic field is varied from −360mT to 360mT and the polarization rotation in transmission is measured (see Fig. 2(b)). Then, the saturated polarization rotation is plotted as a function of the wavelength. The transmittance was measured by a near infrared spectrophotometer. The simulations have been carried out with a homemade RCWA code, taking into account the whole permittivity tensor [30].
3. Results and discussion
The magnetic field $\overrightarrow {B}$ is applied in the x direction referring to Fig. 1(a) (longitudinal configuration) and the incident angle is varied from $0^{\circ }$ to $5^{\circ }$. The incident light is polarized TE (in y direction) or TM (in x direction).
The numerical simulations and experimental measurements of the polarization rotation in transmission $\theta (^\circ )$ and the transmittance T for TE-polarized incident light are plotted on Fig. 3 for the different AOI. By increasing the AOI the resonance wavelength increases, which is consistent with the coupling Eq.(1). One can see dips in transmittance down to 40% for the normal incidence and 60% for the oblique incidence, revealing the waveguide mode resonance. This latter can be identified by the simulated distribution of the electric field intensity $|E_y|$ of TE-polarized incident light illustrated on Fig. 4(a-b), for normal and oblique (AOI=$3^\circ$) incidence at the corresponding coupling wavelengths. For normal incidence, the field amplitude is higher than that of the oblique incidence and the field distribution is not the same. This is related to the fact that for AOI=$0^\circ$, the orders $m=\pm 1$ are exited simultaneously in the MO waveguide, whereas for oblique incidence the two orders are excited at different wavelengths referring to Eq.(1). This behavior explains the difference in the dips amplitude of the transmittance plotted in Fig. 3(a-c), between the normal and oblique incidence.
Fig. 3. The left-hand column displays the measured (a) transmittance and (b) polarization rotation for TE-polarized incident light and for different angles of incident as a function of the incident wavelength. The experimental data agree with the performed simulation shown in the right-hand column (c),(d). The inset figures (a) and (c) represent respectively the experimental measurements and the numerical simulations of the transmittance for TE and TM-polarized incident light at the normal incidence.
Fig. 4. Simulated distribution of the electric field intensity $|E_y|$ of TE-polarized incident light in the x-z plane at the coupling wavelengths (a) for the normal incidence ($\lambda =1530nm$) and (b) for AOI=$3^\circ$ ($\lambda =1580nm$).
The presence of large opposite peaks for a fixed AOI in the polarization rotation spectrum (Fig. 3(b) and Fig. 3(d)) is linked to the transmission resonances for both polarizations (TE and TM). The measured polarization rotations reach $0.5^\circ$ and $-0.85^\circ$ as highest values for the different AOI except the normal incidence. In the latter case and due to symmetry reasons, there is no MO longitudinal effect [31]. The same order of rotation is obtained for TM-polarized incident light (not shown here). Furthermore, a good agreement can be observed between simulations and measurements, and the small difference can be explained by the imperfection of the fabricated structure. We should mention that for a single MO film, the longitudinal effect in transmission is in order of $0.001^\circ$ for small AOI. Hence, a giant enhancement is demonstrated for the MO longitudinal effect at small AOI.
From inset Fig. 3(a), one can observe that the difference between TE and TM resonance’s wavelengths is 13nm. Thus, the optimized condition, where these resonances should overlap for a maximum MO effect, is not satisfied.
However, an overlap can be simply attained through the modification of the AOI. Fig. 5(a-d) represents the simulated diagrams of the transmittance and the polarization rotation in the longitudinal configuration for TE and TM polarized incident light, as a function of the AOI and the incident wavelength. The black and white dashed lines in Fig. 5(a-b) indicate the excitation of the TM and TE waveguide modes respectively. The double dashed lines of each color correspond to the orders $m=\pm 1$ (referring to Eq.(1)). As seen in Fig. 5(a-b), the TE and TM mode dispersion curves (opposite diffraction order sign) crossed at AOI=$0.4^\circ$ for $\lambda =1522.5nm$. According to Eq.(1) it is not a phase matching, the TE and TM modes still have different propagation constants. However, the two modes are excited simultaneously in the MO waveguide resulting in higher MO response. Therefore, as seen in Fig. 5(c-d), the maximum polarization rotation occurs at AOI=$0.4^\circ$.
Fig. 5. Simulated diagrams of transmittance for (a) TE and (b) TM polarized incident light as a function of the wavelength and the angle of incidence. Simulated diagrams of polarization rotations in transmittance for (c) TE and (d) TM polarized incident light as a function of the wavelength and the incident angle. The black and the white dashed lines indicate the excitation of TM and TE waveguide modes respectively.
For experimental verification, Fig. 6(a) depicts the transmittance measurements for TE and TM polarized incident light and the measurements of $\theta (^\circ )$ for TE polarization at AOI=$0.4^\circ$ as a function of the wavelength of the incident light.
Fig. 6. (a) Measurements of transmittance for TE and TM-polarized incident light and measurements of polarization rotation for TE polarization for an angle of incidence equal to $0.4^{\circ }$ as a function of the incident wavelength, (b) Calculated values of figure of merit for the corresponding experimental measurements.
As seen in this figure, the TE and TM modes are excited simultaneously in the MO film as demonstrated in the simulations (Fig. 5(a-b)), and a higher rotation of $1.1^\circ$ is demonstrated at $\lambda =1506nm$ in comparison to the values reached with the other AOI ($\theta =0.85^\circ$ presented in Fig. 3(b)). As well as, two smaller rotations of $-0.43^\circ$ and $-0.5^\circ$ were detected at $\lambda =1493nm$ and $\lambda =1521nm$ respectively.
With the significant demonstrated values of the polarization rotations, this structure is promising for applications such as non-destructive testing based on magnetic field detection or magnetic field sensors, knowing that NPs with smaller diameters must be used in order to have a linear behavior between the rotation and the magnetic field at small values.
For a reliable analysis, a compromise between intensity and rotation should be taken into consideration, through the study of the figure of merit defined as $FoM(^\circ )= \sqrt {T}\times {|\theta (^\circ )|}$ [32]. The calculated values of FoM for experimental measurements are plotted in Fig. 6(b) for TE polarization and for AOI=$0.4^{\circ }$. One can see three peaks in the FoM curve corresponding to the three MO resonances in Fig. 6(a). High values of FoM are reached and the more important one is $0.83^\circ$ at $\lambda =1506nm$.
This value is higher than that obtained by Chetvertukhin et al. [33], who experimentally demonstrated a value of $0.018^\circ$ for the longitudinal effect but in reflection. The work was done with a magnetoplasmonic crystal consisting of a 2D array of nickel nanodisks arranged into hexagonal lattice. The off-diagonal element of the tensor permittivity of the nickel is around: $\epsilon _{23}=+i0.24$ [34]. Kalish and Belotelov [11], have numerically demonstrated a value of FoM equal to $2.53^\circ$ for the longitudinal effect in transmission with a magneto-plasmonic structure formed by a gold grating deposited on a magnetic layer of rare-earth iron garnet containing bismuth. Here, the off-diagonal element is: $\epsilon _{23}=+i0.016$ [8]. Hence, same order of FoM was demonstrated in our work with a MO material owing off-diagonal element ($\epsilon _{23}=-i0.0064$) in one or two orders of magnitude smaller as compared to these works.
4. Conclusion
Large enhancement of the longitudinal MO effect accompanied with high transmission was experimentally and numerically demonstrated for small angles of incidence. The fabricated structure consists of a dielectric grating deposited on the top of a MO composite waveguide made of magnetic NPs embedded in a silica matrix. The whole permittivity tensor of the MO composite can be tuned by modifying the volume fraction of the NPs. Finally, the simplicity of fabrication and the significant polarization rotation of the structure open the field of applications in magnetic field sensors or in non-destructive testing.
Funding
Agence Nationale de la Recherche (ANR-16-CE09-0017).
Disclosures
The authors declare no conflicts of interest.
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