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Abstract
A magnetophotonic Tamm structure with a bilayer film of Bi-substituted iron garnets (Bi:IG) with a significant magneto-optical (MO) response, placed in microcavity, and Tamm plasmon-polaritons (TPP) excited at the interface of the Au layer, were proposed. Optical and MO spectra of the hybrid state were investigated theoretically and experimentally in Faraday effect geometry at a normal incidence of the light wave. Anticrossing behavior of the TPP and Fabry-Perot resonances and dependence of spectral splitting on their hybrid state on the thickness of the Au layer was shown. By varying the thickness of the top SiO2 layer in the structure, the modifications of spectra on the TPP spectral position were described in detail. The results can be used to design tunable MO devices and sensors.
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1. Introduction
Magnetophotonic structures with propagating surface electromagnetic waves (SEW), non-propagating surface states, surface plasmon-polaritons (SPP), localized surface plasmons (LSP) and their various hybrid states are interesting from both fundamental and technological standpoints due to their unique properties applied in magnetic field sensors, optical biosensors, optical isolators, data storage and fast optical modulation [1–7]. Optical Tamm state (OTS) is a type of non-propagating surface states. The term "Tamm state" was borrowed from solid state physics [8] to the terminology of photonics to describe SEW excited at the boundary of photonic crystal (PC) [1–3,9–15]. OTS arises between two different PCs with overlapping photonic band gaps (PBG) or between a PC and an isotropic medium with negative permittivity ($\epsilon < 0$). In the second case, a coupled mode of standing SEW and surface plasmon excitation is formed, so-called TPP [10]. TPP has many similarities with SPP and SEW in PC [2,3,9–15]. Nevertheless, TPP is excited by TM and TE incident wave in contrast to SPP. TPP excitation does not require prism or diffraction methods. The first experimental observation of TPP is described in [10]. The influence of TPP excitation on the properties of magnetophotonic crystals, PC with magnetic layers, is reported in [1–3]. Such magnetophotonic crystals are called "Tamm structures" [1]. The structures distinguish by the ability to control their optical properties by magneto-optical (MO) effects. It becomes possible to modulate the intensity, phase and polarization of optical waves by magnetic fields with a frequency of several tens or even hundreds of GHz. The intensity of MO effects in PC enhances by interference, diffraction and localization of light [16]. Light localization causes of many attractive effects in the majority of optoelectronic components, for example, even in optical fibers [17,18]. In the works [1,19] Tamm structure, synthesized on the basis of Bi-substituted iron garnet (Bi:IG) magnetic films transparent for the visible and near infrared ranges and Au plasmonic layer, was first introduced.
Simultaneous excitation of several resonances, i.e. hybrid states of TPP and resonances of other nature [19–21], allows to modify the dispersion laws of independent excitations. The presence of hybridization gives additional possibilities to control spectral position and resonance amplitude of modes. Thus, the investigations [13,20] are devoted to hybrid state of TPP and Fabry-Perot (FP) mode. In [14,19] the properties of hybrid state of TPP and SPP are considered. Resonance crossing has been proposed to increase the sensitivity of optical biosensors [22]. Simultaneous formation of TPP and localized state similar to the defect mode inside the dielectric nanocomposite layer with metallic nanoparticles was demonstrated also in [21].
In our previous works [23–26] new original Tamm structures with Bi:IG and Au plasmonic layers were proposed. The structures with single and double garnet layers were modelled to form a TPP mode at the center of PBG [26]. Tamm structures were constructed on the basis of dielectric non-magnetic Bragg mirrors, which are sequentially coated with layers of Bi:IG, $\mathrm {SiO_2}$ and Au. In this case, TPP resonance corresponds to the localization of electromagnetic field of light wave inside the layers adjacent to Au, that is the buffer $\mathrm {SiO_2}$ and Bi:IG layers. Subsequently we suggested the MO microcavity coated by Au layer with thickness gradient [25]. The features of TPP resonances as a function of Au thickness have been found. However, since the spectral position of TPP in the previously considered structures was fixed, the influence of simultaneous excitation of TPP and FP mode inside PBG on optical and MO properties of magnetophotonic structures has not been previously demonstrated theoretically and experimentally. As shown in [27], TPP position (wavelength or frequency) is varied by the top layers thickness. In MO microcavity light is localized inside the cavity layer that leads to a significant increase in transmittance and MO Faraday effect. The amplification factor of Faraday effect of MO microcavity relative to the used Bi:IG layers reaches 50-60 times at values of transmittance from 14$\%$ to 40$\%$ at the resonant wavelength. Thus, the structures provide a high MO quality factor for various applications [23,24,28]. Magnetophotonic Tamm structures [1,25,26] due to the presence of Au layer with a sufficiently high optical absorption, have less attractive characteristics. The maximum achieved enhancement of Faraday effect is 10 times at values of transmittance less than 4$\%$ at the resonant wavelength. In considered structure we use a Bi:IG cavity and a top buffer layer of $\mathrm {SiO_2}$ with gradient of thickness to control the TPP spectral position. As will be demonstrated below, a distinctive feature of proposed Tamm structure is that the spectral proximity of TPP and FP modes leads, on the one hand, to an increase in transmittance of TPP resonance and, on the other hand, allows one to control the degree of localization, amplification of MO effects and spectral position of FP mode.
2. Structure and methods
2.1 Model and simulation
We modeled, synthesized and experimentally investigated the structure with configuration $\mathrm {GGG/[TiO_{2}/SiO_{2}]^{4}/M1/M2/[SiO_{2}/TiO_{2}]^{4}/SiO_{2}/Au}$. There GGG is substrate of gadolinium gallium garnet (GGG) of (111) crystallographic orientation. $\mathrm {[SiO_{2}/TiO_{2}]^{4}}$ and $\mathrm {[TiO_{2}/SiO_{2}]^{4}}$ are non-magnetic Bragg mirrors with four pairs of layers of titanium $\mathrm {TiO_2}$ and silicon $\mathrm {SiO_2}$ oxides. M1 and M2 are Bi:IG films of compositions $\mathrm {Bi_{1.0}Lu_{0.5}Gd_{1.5}Fe_{4.2}Al_{0.8}O_{12}}$ and $\mathrm {Bi_{2.8}Y_{0.2}Fe_{5}O_{12}}$, respectively. $\mathrm {SiO_{2}/Au}$ are top silicon oxide (buffer $\mathrm {SiO_{2}}$) and gold layers. To form Tamm structure, MO microcavity with the bilayer Bi:IG film was used as a one-dimensional magnetophotonic crystal. MO microcavity configuration was proposed earlier and its design and synthesis procedure are described in detail in [23,24]. Bilayer film provides a significant increase of the MO rotation and quality factor of microcavity due to the technology of two-stage synthesis of garnets with different Bi contents. Considered structure with top $\mathrm {SiO_{2}}$ layer of gradient of thickness $\mathit {h}_{\mathrm {bSiO}_2}$ is shown in Fig. 1 (a).
Fig. 1. Schematic representation of considered structure with beam path diagrams illustrated the FP and TPP excitation (a); measured (symbols) and simulated (lines) spectra of $K_{\mathrm {t}}$ and $\theta _{\mathrm {F}}$ of MO microcavity (b).
Modeling and evaluation of structure parameters by fitting of measured optical and MO spectra were carried out by transfer matrix method [30,31]. The method consists in the numerical solution of Maxwell’s equations using fourth-order matrices, which describe the change in the state of electromagnetic wave passing through a separate layer of a given thickness (or the structure as a whole). Maxwell’s equations for considered media in the optical range are written as
Components of permittivity tensors of layers were defined earlier using the experimental data of single films and synthesized MO microcavity [24,29]. The permittivity tensor for Bi:IG layers has the form
Index j is "TiO2" for $\mathrm {TiO_2}$ layer, "SiO2" for $\mathrm {SiO_2}$ layer and "Au" for Au layer.
The dispersion of components of permittivity tensors of layers are listed in [24,29]. Here we give the values for $\lambda _{\mathrm {R}}=657$ nm. The permittivity tensor components of MO active layers are $\epsilon _\mathrm {xxM1}=\mathrm {6.540+0.110}\cdot i$, $g_\mathrm {M1}=-\mathrm {0.009+0.003}\cdot i$ for M1 and $\epsilon _\mathrm {xxM2}=\mathrm {7.807+ 0.111}\cdot i$, $g_\mathrm {M2}=-\mathrm {0.041+0.007}\cdot i$ for M2. The permittivity tensor components of layers $\mathrm {SiO_2}$, $\mathrm {TiO_2}$, Au and substrate GGG are chosen as $\epsilon _\mathrm {SiO2}$=2.120, $\epsilon _\mathrm {TiO2}$= 5.266, $\epsilon _\mathrm {Au}=-\mathrm {13.370+1.178}\cdot i$ and $\epsilon _\mathrm {GGG}$= 3.859, respectively.
We considered that TM polarized light falls on the structure; the structure has flat smooth interfaces of layers and is characterized by the absence of thickness gradient of layers in all simulations. The algorithm presented in [30,31] is implemented for numerical calculations of components of electric fields vectors of light wave $E^{\mathrm {TM}}$ and $E^{\mathrm {TE}}$. Accordingly, Faraday rotation angle and transmittance are determined as
To optimize the configuration of Tamm structure for synthesis, we have changed the thickness of top $\mathrm {SiO_2}$ and Au layers.
2.2 Sample fabrication and characterization of layer parameters
The layers of $\mathrm {TiO_2}$ and $\mathrm {SiO_2}$ of Bragg mirrors were synthesized by electron beam evaporation on hot (400$^{\circ }$C) substrate. The thicknesses of layers were optically controlled in situ. Bi:IG layers of structure were fabricated by reactive ion beam sputtering of corresponding ceramic targets in argon-oxygen mixture on "cold substrates" (80$^{\circ }$C) and crystallization in air at atmospheric pressure. The synthesis of bilayer film consisted of two stages: the deposition and annealing at temperature $T_a = 700^{\circ }$C M1 single film and the deposition and annealing at $T_a = 680^{\circ }$C M2 single film. The duration of each of annealing processes was 20 min. Proposed synthesis method [24–26,29] of a bilayer film allows to eliminate the problem of growth of garnet phase with a high Bi content on $\mathrm {SiO_{2}}$ layer, as its occurs by mechanism of spontaneous crystallization. In this case, the first layer M1 acts as a sublayer, which stimulates the growth of a phase with a high Bi concentration M2. According to our previous investigations [23,24], the formation of a single M2 film on $\mathrm {SiO_{2}}$ layer does not occur. Thus, the bilayer film made it possible to increase the specific $\theta _\mathrm {F}$ of MO active cavity in magnetophotonic crystals at 655 nm from 0.9$^{\circ }/\mu$m (typical for M1) to 3.6$^{\circ }/\mu$m (typical for M1/M2 film) [23,24].
The top $\mathrm {SiO_{2}}$ layer placed between MO microcavity and Au coating was formed with gradient of thickness by reactive ion-beam sputtering. The method of a "thin shutter" was applied to form the thickness gradient [32,33]. It was implemented technically by creating a "penumbra" region during the deposition of target material onto a substrate. The "thin shutter" with micrometrically sharpened edge was placed between the target bombarded by ion source and the substrate. Therefore, the three areas in the plane of substrate existed. The first one was the area with maximum thickness almost unchanged within the site. The second one was the middle zone of geometric "penumbra", in which the deposited layer had been monotonously decreasing in thickness and formed the "wedge". The third one was the area that had been free from deposited material. The width of second "penumbra" area is determined by target diameter, the distances from target to thin shutter and from target to substrate. In the setup configuration, these parameters were selected in such a way that the formation of a wedge occurred on most of the substrate surface along its long side. This technology was proposed for the first time to synthesize a dielectric layer of a magnetophotonic crystal. According to our measurements and simulations, the thickness $\mathit {h}_\mathrm {bSiO_2}$ varies from 120 to 230 nm along a side of the sample with the length of 12 mm. So, the value of gradient is 9 nm/mm.
The top Au thin film was deposited by thermal evaporation in vacuum. Thickness of top Au layer $\mathit {h}_\mathrm {Au}$ was chosen equal to 40 nm, since at this thickness sharp TPP resonance of the highest optical quality factor can be formed [26].
The thickness of layers of Tamm structure during the synthesis was controlled by sputtering time and deposition rate. Thin layers of $\mathrm {SiO_{2}}$, $\mathrm {TiO_{2}}$, M1 and M2 have been previously investigated using optical microinterferometry (MII-4) and atomic force microscopy (SPM NTEGRA, NT-MDT). The error in the thickness of deposited layers did not exceed $10\%$. Semicontact atomic force microscopy (AFM) was applied to investigate the morphology of layers. The measurements were carried out by cantilevers of HA-HR ETALON.
2.3 Optical and magneto-optical measurements
Spectra of transmittance $K_\mathrm {t}$ were measured by automated spectrophotometer KFK-3 at wavelength range from 400 to 990 nm. The transmittance of the sample was determined by ratio of luminous fluxes of the light incident on the sample $F_\mathrm {0}$ and the light transmitted through the sample $F_\mathrm {t}$. In the scheme of spectrophotometer the light emitted by halogen lamp and passed through a monochromator with diffraction grating and output slit, interacts with the sample and then detected by photodetector. Luminous flux $F_\mathrm {0}$ and $F_\mathrm {t}$ is converted by photodetector into electrical signals $U_\mathrm {0}$ and $U_\mathrm {t}$, respectively. So, the transmittance $K_\mathrm {t}$ is expressed as
Here $U_\mathrm {N}$ is a signal without illumination.
The spectral interval allocated by the monochromator is no more than 1.5 nm. The limit of the permissible value of absolute error of the measurements is $\pm 2\%$. The minimum detectable value of transmittance $K_\mathrm {t}$ is 0.1$\%$.
Investigation of spectral dependences of Faraday rotation angle $\theta _\mathrm {F}$ was carried out using handmade automated spectropolarimeter by compensation method in field $H =$ 2 kOe that exceeds the value of saturation field of Bi:IG bilayer in Faraday geometry ($H_\mathrm {S}$ = 1.6 kOe). Beam aperture and gradient of $h_\mathrm {bSiO_2}$ at aperture scale were 0.1 mm and 1 nm, respectively. The block-scheme of the spectropolarimeter contains the same units as the spectrophotometer, only two Glan-Taylor prism placed before and after the sample located in the hole of electromagnet are included to realize the compensation by rotating analyzer. The spotting of minimal values of transmittance, when analyzer position is fixing and polarizer position is rotating, was implemented to fix the rotation angle of polarization plane of light. To exclude the nonmagneto-optical contribution to rotation angle of light polarization, the Faraday rotation $\theta _\mathrm {F}$ for each wavelength were determined as half-difference of two values $\theta (+H)$ and $\theta (-H)$ obtained for two opposite directions of magnetic field of the same strength, respectively:
Faraday rotation measurement range of the spectropolarimeter is $\pm 90^{\circ }$ with precision of $0.01^{\circ }$ at transmittance above $20\%$.
3. Results and discussions
Figure 1 (b) illustrates the spectra of $K_{\mathrm {t}}$ and $\theta _\mathrm {F}$ of MO microcavity $\mathrm {GGG/[TiO_2/SiO_2]^{4}/M1/M2/}$ $\mathrm {/[SiO_2/TiO_2]^{4}}$. As seen, PBG is located between 580 and 825 nm. The resonant wavelength of FP mode is $\lambda _\mathrm {R}=657$ nm. The thicknesses of layers $\mathrm {TiO_2}$, $\mathrm {SiO_2}$, MO sublayer M1 and main MO layer M2 are $h_\mathrm {TiO_2}=73$ nm, $h_\mathrm {SiO_2}=115$ nm, $h_\mathrm {M1}=66$ nm and $h_\mathrm {M2}=166$ nm, respectively. Optical thickness of Bi:IG cavity corresponds to second-order condition of FP resonance and is equal to $\lambda _\mathrm {R}$. Simulation of Tamm structure properties before synthesis was performed based on the layers parameters of MO microcavity changing the thicknesses of top layers $h_\mathrm {bSiO_2}$ from 110 to 320 nm and $h_\mathrm {Au}$ from 0 to 70 nm. Obtained spectra of $K_\mathrm {t}$ and $\theta _\mathrm {F}$ as a function of thicknesses $h_\mathrm {bSiO_2}$ and $h_\mathrm {Au}$ are shown in Figs. 2 and 3.
Fig. 2. Image plots of spectra of $K_\mathrm {t}$ (a, c, e, g) and $\theta _\mathrm {F}$ (b, d, f, h) of Tamm structure with $h_\mathrm {Au}$ of 10 nm (a, b), 20 nm (c, d), 40 nm (e, f) and 70 nm (g, h) as a function of $h_\mathrm {bSiO_2}$. The upper graphs in figures show the spectra corresponding to the cross sections of the same color in the plots. Dependencies of resonant wavelengths $\lambda _\mathrm {R}$ and $\lambda _\mathrm {TPP}$ as a function of $h_\mathrm {bSiO_2}$ (i, j, k, l).
As evident from the plots, behavior of amplitudes and positions of FP and TPP resonances are substantially modified relative to the case of independent excitation of FP and TPP [24,26] and significantly depends on relative position of resonances inside PBG and thicknesses $h_\mathrm {bSiO_2}$ and $h_\mathrm {Au}$.
The shift of TPP mode from shortwave to longwave edge of PBG is caused by varying $h_\mathrm {bSiO_2}$ from 110 to 330 nm. Thickness $h_\mathrm {Au}$ strongly affects the characteristics of hybrid state. The most coupled states are observed for configurations with the closest spectral location of TPP and FP resonances at $h_\mathrm {bSiO_2}$=155 nm ($h_\mathrm {Au}$=10 nm), $h_\mathrm {bSiO_2}$=175 nm ($h_\mathrm {Au}$=20 nm), $h_\mathrm {bSiO_2}$=184 nm ($h_\mathrm {Au}$=40 nm) and $h_\mathrm {bSiO_2}$=185 nm ($h_\mathrm {Au}$=70 nm). These cases correspond to resonance crossing positions. It is apparent that spectral splitting $\Delta \lambda =| \lambda _\mathrm {R} - \lambda _\mathrm {TPP}|$ of resonances exists and changes with increasing of $h_\mathrm {Au}$. The value of $\Delta \lambda$ is 0 for $h_\mathrm {Au}$=10 nm and 17 nm for $h_\mathrm {Au}$=40 nm. So, a resonance or two resonances arise. It is evident also the shift of FP resonance caused by the coupling to TPP (Fig. 3). Spectral splitting of resonances of hybrid state occurs in the range of thickness $h_\mathrm {Au}$ from 20 to 30 nm and is associated with increasing of optical figure of merit of TPP [26].
Fig. 3. Image plots of spectra of $K_\mathrm {t}$ (a) and $\theta _\mathrm {F}$ (b) of Tamm structure with $h_\mathrm {bSiO_2}$=180 nm as a function of $h_\mathrm {Au}$. The upper graphs in figures show the spectra corresponding to the cross sections of the same color in the plots.
It should be noted two features of hybrid state. The first one, resonant value of $K_\mathrm {t}$ of hybrid state at resonance crossing is significantly higher than resonant values of $K_\mathrm {t}$ of a TPP or a FP peaks without resonance crossing at the same $h_\mathrm {Au}$. The second one, the value of $\theta _\mathrm {F}$ of hybrid state in the case when TPP resonance shifts to a wavelength with FP resonance is smaller in twice compared to the values of $\theta _\mathrm {F}$ in other cases.
Based on modelling we synthesized the structure with $h_\mathrm {Au}$=40 nm and measured spectra of structure at different positions of thickness gradient of top $\mathrm {SiO_{2}}$ layer.
Measured and calculated optical and MO spectra are shown in Fig. 4.
Fig. 4. Measured (symbols) and calculated (lines) spectra of $K_\mathrm {t}$ (a) and $\theta _\mathrm {F}$ (b) of synthesized Tamm structure with $h_\mathrm {Au}$=40 nm as a function of $h_\mathrm {bSiO_2}$. The values of $K_\mathrm {t}$ for calculated spectra are reduced by 5 times. The values $h_\mathrm {bSiO_2}$ are the same for (a) and (b) graphs. Figure (c) shows the comparison of measured and calculated spectra of $\theta _\mathrm {F}$ of configurations with $h_\mathrm {bSiO_2}$ 129 nm and 185 nm.
In experimental spectra the broadening of resonance FP peak are obvious, and the transmittance of TPP and FP modes is attenuated with respect to numerical values (more than in 5 times). Partially, the limitations of the modelled spectra can be related to granularity, roughness and the anomalous dispersion in the shortwave spectral region of optical constants of Bi:IG films [24,34]. These factors difficult to take into account in theory. The calculations took into account the monotonic change in the components of dielectric tensor, which agreed with the experimental data for MO microcavity in the range above 580 nm (Fig. 1, b). The formation of FP and TPP resonances in the spectra of Tamm structure occurs in this range. According to AFM measurements, the microstructure of Bi:IG bilayer M1/M2 is polycrystalline with an average grain size (AGS) of 125 nm and a root mean square roughness (RMS) of the surface of 7 nm. For M1 single layer the same parameters are AGS = 90 nm and RMS = 5.5 nm [35]. So, an additional absorption and scattering from real-structure defects and rough surfaces can significantly affect in this case. In addition, in the experimental spectra TPP resonance is broadened significantly and its amlitude is reduced relative to the calculated lines. This fact is associated with the presence of a gradient of $h_\mathrm {bSiO_2}$ in the beam aperture during measurements and the roughness of Bi:IG bilayer, top buffer $\mathrm {SiO_{2}}$ and Au layers. $\mathrm {TiO_{2}}$ and $\mathrm {SiO_{2}}$ layers have a smooth surface with RMS of less than 4 nm. Similar to garnet films, the Au layer is formed by a polycrystalline with AGS of 85 nm and RMS of 4 nm. This may be the main reason for the broadening of TPP peak and a significant decrease in transmittance relative to the model spectra, since the quality of metal coating strongly affects the TPP excitation at the layer interface. Nevertheless, all these factors are unaffected the experimental observation of main features of mode interaction. Changes of resonance values of $K_\mathrm {t}$ and $\theta _\mathrm {F}$ and positions of modes are confirmed by the experimental data. Faraday rotation $\theta _\mathrm {F}$ of FP resonance of Tamm structure, when resonance crossing does not take place and resonances do not aligned at a wavelength, is higher of the same value of a MO microcavity more than $10\%$.
This MO enhancement is also associated with hybridization of modes that presents in all configurations of proposed Tamm structure. The results on the enhancement of $\theta _\mathrm {F}$ for this structure are lower than previously proposed for magnetoplasmonic structure [36]. Nonetheless, the amplification of Faraday effect the order from 8 to 17.5 times depending on the $h_\mathrm {bSiO_2}$ is present with respect to the case of a Bi:IG bilayer. In addition, the structure allows control and amplification of $\theta _\mathrm {F}$ not only by external magnetic fields.
To clarify behavior of hybrid mode, we calculated the spatial distribution of electric field intensity $I$ inside the structure for FB and TPP resonant wavelengths (see Fig. 4). The results of simulation are shown in Fig. 5 (a and b).
Fig. 5. Calculated spatial distribution of electric field intensity at TPP (a) and FP (b) resonant wavelengths inside of Tamm structure with $h_\mathrm {Au}$=40 nm and $h_\mathrm {bSiO_2}$ of 220 nm, 185 nm and 147 nm. The bottom graph in (b) shows the distribution inside MO microcavity. The values $h_\mathrm {bSiO_2}$ are the same for (a) and (b) illustrations on the same level. Resonant wavelengths (c) and MO quality factor (d) of TPP and FP peaks for the structure as a function of $h_\mathrm {bSiO_2}$.
Distribution of real part of diagonal component of permittivity tensor $\mathrm {Re}(\epsilon )$ inside the structure is also presented at the top of each chart. It was assumed that light falls normal to the structure surface on a side of Au coating. The substrate is not taken into account in the calculation.
Hybridization of two different states implies the formation of a mixed state that simultaneously possesses the characteristics inherent in each of them separately. The intensity for TPP mode increases dramatically to interface of "top $\mathrm {SiO_{2}}$ layer – Au layer" and its maximum is inside $\mathrm {SiO_{2}}$ layer [1,24,26] (Fig. 5, a; $h_\mathrm {bSiO_2}$ = 220 nm). FP mode is characterized by localization of electric field at the vicinity of interfaces or inside of magnetic layers (Fig. 5, b; $h_\mathrm {bSiO_2}$ = 220 nm). For configuration with $h_\mathrm {bSiO_2}$ = 185 nm the intensity distribution for both modes is similar and indicates the strongest hybridization (Fig. 5, a and b). Electric field is localized inside or at the vicinity of magnetic and top $\mathrm {SiO_{2}}$ layers simultaneously, while maximum of intensity is located inside the layer adjacent to the Au layer. The electromagnetic wave is localized in two different regions of the crystal simultaneously. Therefore, the resonances exhibit anticrossing behavior in the spectra (Figs. 2, 3, 4), and a TPP resonance cannot be combined at a wavelength with a FP resonance in the cases when TPP resonance is amplified by thickening of metal layer (at $h_\mathrm {Au}$ of more than 20 nm). According to calculations, despite the two peaks are separating in the spectra, their characteristics are identical (intensity distribution in the structure, $K_\mathrm {t}$, $\theta _\mathrm {F}$) when they are crossing.
Any reconstruction in the structure leading to a change in the optical figure of merit of states [37], i.e. to a redistribution of the intensity maxima. So, in the case of an increase in the optical figure of merit of TPP resonance with the growth of thickness of Au layer, the localization of light in the top $\mathrm {SiO_{2}}$ layer increases and decreases in Bi:IG bilayer cavity. This fact leads to a change in the resonance conditions of the beams multiple reflected inside the structure and, as a consequence, to a change in the positions of resonances when they crossing (Fig. 2). FP mode becomes more controllable due to TPP excitation. On the other hand, when we change the position of TPP resonance by varying the thickness of top $\mathrm {SiO_{2}}$ layer, a redistribution of the intensity inside the crystal occurs. It should be noted that a distinctive feature of the structure under discussion is the change in the values of $\theta _\mathrm {F}$ of resonances, the ratio of which can be considered (only in our case) as the measure of hybridization:
The maximum contribution to $\theta _\mathrm {F}$ gives the MO active layer in which the greatest amplification of electromagnetic field occurs. When the resonances are crossing, the TPP pulls the intensity distribution into the region of top $\mathrm {SiO_{2}}$ layer, decreasing the contribution to $\theta _\mathrm {F}$ of Bi:IG bilayer cavity ($\theta _\mathrm {F}$ of FP resonance). At the same time, as the resonances are crossing, the values of $\theta _\mathrm {F}$ for TPP resonance increase. This is explained by the fact that upon reaching the resonance conditions inside Bi:IG bilayer cavity, the formation of maximum intensity of light wave field occurs due to the excitation of TPP state itself and multiple reflection of the beams from the Au layer. So, for configurations with $h_\mathrm {bSiO_2}$ = 185 nm and $h_\mathrm {bSiO_2}$ = 220 nm $r$ is 0.68 and 0.025, respectively.
As it seen from Fig. 5 (c, d), hybridization influences on values of MO quality factor [28] of resonances determined as
Obtained values of $Q$ are higher than $Q$ of the first magnetophotonic Tamm structure $[\mathrm{SiO_{2}}/ \mathrm{Bi:IG}]^{5}/\mathrm{Au}$ $(Q=0.58{{}^{\circ }})$ [1]. However, values are lower than the MO quality factor of Tamm structures $\mathrm {GGG/[TiO_{2}/SiO_{2}]^{7}/M1/M2/SiO_{2}/Au}$ proposed us in [26] ($Q$ is varied from 0.55 to 5$^{\circ }$ at Au thickness changes from 0 to 65 nm). We expect that MO quality factor of magnetophotonic Tamm structure with hybrid state might be higher significantly. To optimize the optical and MO parameters of the structure, it is possible to vary other parameters, the influence of which was not discussed in the investigation, such as: the number of layer pairs in Bragg mirrors $m$; thickness of Bi:IG bilayer cavity; symmetry of location of Bi:IG bilayer cavity in photonic crystal (non-symmetric MO microcavity).
4. Conclusion
We demonstrated the original magnetophotonic Tamm structure with hybrid state of TPP and FP. Proposed structure was formed on bilayer film of Bi:IG of composition $\mathrm {Bi_{1.0}Lu_{0.5}Gd_{1.5}Fe_{4.2}} \mathrm{Al_{0.8}O_{12}/}$ $\mathrm {Bi_{2.8}Y_{0.2}Fe_{5}O_{12}}$, placed in microcavity, and plasmonic Au layer.
It was found that hybrid state of TPP and FP has anticrossing behavior. Simulation of spectral properties of the structure with different parameters of layers shows that spectral splitting of resonances depends on the thickness of Au layer $h_\mathrm {Au}$. Splitting does not occur in the case when $h_\mathrm {Au}$ does not provide the TPP resonance with a sufficiently high optical figure of merit ($h_\mathrm {Au}$ $\leqslant$ 25 nm).
Simulations and experimental investigations of the structure with high degree of hybridization, when resonance crossing takes place, show that $K_\mathrm {t}$ of hybrid mode increases at least in 4-6 times over the other cases. As this takes place, $\theta _\mathrm {F}$ of FP resonance decreases in 1.5-2 times and spectral shift of FP resonance wavelength arises. Maximum shift is 14 nm at $h_\mathrm {Au}>30~$nm.
Experimental dependences are show that the presence of hybrid state of TPP and FP mode allows one to control MO quality factor and other parameters of resonances by the thickness of top non-magnetic $\mathrm {SiO_{2}}$ layer of the structure. These features of proposed structure can be taken into account in design of tunable MO devices and optical sensors.
Funding
Ministry of Science and Higher Education of the Russian Federation (075-15-2019-1934).
Disclosures
The authors declare that there are no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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