1. Introduction

Physics of hyperbolic metamaterials (HMM) is nowadays one of the most popular branches of nanophotonics as such materials exhibit unique properties perspective for applications. HMM are artificial strongly anisotropic uniaxial metal-dielectric nanostructured media with opposite in sign principal components of the permittivity and/or permeability tensor [1–3], which results in their hyperbolic dispersion law instead of elliptical one intrinsic for traditional optical materials. If $\mu =1$, the isofrequency HMM surface for the TM mode is given by $\frac {k_x^2+k_y^2}{\epsilon _\parallel }+\frac {k_z^2}{\epsilon _\perp }=\left (\frac {\omega }{c}\right )^2$, where $\omega$, $k_{x,\;y,\;z}$ are the frequency and related components of the wave vector of the electromagnetic wave, $c$ is the speed of light; the dielectric tensor components describing the permittivity along and perpendicular to the optical axis, $\epsilon _\parallel$ and $\epsilon _\perp$. The two simplest optical HMM structures are the multilayered films made of alternating metal/dielectric nanolayers, and arrays of metal nanorods in dielectric templates. Uniaxial anisotropy of HMM is responsible for the two spectral features of $\epsilon$, the first one is the appearance of zero value of the real part of one of $\epsilon$ components (Epsilon Near Zero, ENZ), the second is the $\epsilon$ pole (Epsilon Near Pole, ENP) associated with the resonant enhancement of the dielectric permittivity and often related to the transversal and longitudional plasmon resonanses in nanorods [4]. Along with the specific dispersion, these features govern a number of interesting optical phenomena including the negative refraction [5], enhanced spontaneous emission [6] and Raman scattering. High potential of the HMM for nanophotonics, imaging [7], and waveguiding [8] is also well recognized. Recently new perspectives for using HMM as optically active devices, namely as functional polarization converters based on an ultrathin plasmonic nanorod metamaterial slab in the vicinity of the ENZ point, have been demonstrated [9].

Additional functional properties can arise if a structure contains a gyrotropic material, thus allowing magnetic field control over its effective permittivity [10]. As has been shown, composite magneto-plasmonic structures of different design can enhance magneto-optical effects close to the resonant spectral points [11–14], whereas similar experiments have not been performed for HMM in the region of the hyperbolic dispersion and at the transition through the ENZ point.

Here we study both experimentally and by modeling the effects of giant polarization plane rotation and ellipticity changes in (i) nonmagnetic HMM made of gold nanorods in porous alumina and (ii) the similar structure with a thin Ni film deposited above the Au nanorods (“HMM$+$Ni film”) in wide angular and wavelength ranges including ENP and ENZ points, as well as the hyperbolic region at $\lambda >810$ nm.

2. Samples and experimental setup

(i) Arrays of gold nanorods in 50 $\mu$m thick anodic aluminum oxide (AAO) matrix were made by templated electrodeposition as described elsewhere [15]. According to the scanning electron microscopy, the composed structures consist of Au nanorods of 40 $\pm$ 4 nm in diameter, which coincides with the pore diameter of the AAO template. The length of the nanorods is 580 $\pm$ 40 nm, and the volume fraction of gold in the layer of AAO template filled by the metal is about 8%, the optical axis being parallel to the Z direction. (ii) the same HMM as (i), with a continuous 15 $\pm$ 3 nm thick Ni layer deposited by DC magnetron sputtering. It was shown previously [15] that the properties of the permittivity of the HMM remain unchanged, as the Ni layer is homogeneous and very thin (15 nm in comparison with 580 nm thickness of the AAO with Au rods).

Transmission spectra were measured using a halogen lamp as a broadband light source, (XOZ) being the plane of incidence and orientation of the polarization plane of the probe beam is determined by the angle $\phi$ counted off from the (XOZ) (see Fig. 1(a)). The polarization state of the transmitted beam was estimated using the ellipsometer (WVASE by J.A. Wollam Co., Inc.).

 

Fig. 1. (a) Scheme of the “HMM$+$Ni film” sample and the experimental setup for the Faraday effect measurements; (b) transmission spectra of the sample for different angles of incidence; (c) calculated spectra of the effective components Re$\epsilon _{\perp }$ (black line), Re$\epsilon _{\parallel }$ (red line). Shaded areas in (b), (c) correspond to the hyperbolic dispersion spectral regions.

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Magneto-optical experiments for the “HMM$+$Ni film” were performed as a DC magnetic field H of 1 kOe formed by two permanent ring magnets was applied in the Faraday geometry (Fig. 1(a)). The Faraday effect is measured via the intensity modulation of the radiation transmitted through the “HMM$+$Ni film” and the analyzer set at 45$^{\circ }$ with respect to the p-polarization; the effect is characterized by the magnetic contrast $\rho =\frac {T(H)-T(-H)}{T(H)+T(-H)}$, where $T(\pm H)$ is the trasmission for the opposite directions of the magnetic field. As a reference magnetic sample without hyperbolic dispersion, we used an AAO with Ni film without nanorods; in that case the ellipticity of the transmitted light can be neglected as the birefringence is weak, and the polarization plane rotation can be estimated as $\alpha =1/2 \arcsin (\rho )$.

3. Experimental results

1. Transmission spectra of the sample (i) for the p-polarized fundamental beam ($\phi$=0) shown in Fig. 1(b) are qualitatively similar to those discussed in [15]. At normal incidence, the resonance associated with the transversal oscillations of the free electron gas in Au nanorods results in a single transmission minimum centered at approximately 540 nm. At oblique incidence (20$^{\circ }$ and 40$^{\circ }$), the minimum at about 810 nm appears caused by the longitudinal plasmon resonance excitation. As expected, it vanishes for the s-polarized incident beam.

Figure 1(c) shows the simulated spectra of the real parts of the effective permittivity components $\epsilon _{\perp }=\epsilon _{xx}=\epsilon _{yy}$ and $\epsilon _{\parallel }=\epsilon _{zz}$ (the coordinate frame is shown in Fig. 1(a)) within the effective medium model [16]. For calculations the parameters of the experimental samples were taken, Au and AAO dielectric constants were taken from [17]. Transmission minima attained at $\lambda =$540 nm and 810 nm are close to the ENP and ENZ points, respectively. Thus the hyperbolic regime corresponds to the range $\lambda >$810 nm shown by grey color in Figs. 1(b) and 1(c).

2. The ellipsometry data are the three complex numbers $T_1, T_2$, and $T_3$ acquired by using the anisotropy measurement options [18]. These quantities are the ratios of the sample’s Jones matrix components normalized by the complex field transmission coefficient $T_{ss}$ for s-polarized light: $J=\begin {pmatrix} T_{pp} & T_{sp} \\ T_{ps} & T_{ss} \end {pmatrix}=T_{ss}\begin {pmatrix} T_1 & T_2 \\ T_3 & 1 \end {pmatrix}$, where $T_1$=tan$\Psi e^{i\Delta }$ is the main ratio equivalent to the standard ellipsometry measurement values. Figure 2(a) shows the obtained $T_1$ polar plot with the ENP and ENZ points indicated for clarity. One can see that the smallest absolute value of $T_1$ corresponds to the ENZ point characterized by a strong absorption of the p-polarized light.

 

Fig. 2. The data obtained from the ellipsometry measurements: (a) the complex Jones matrix element T$_1$ of the transmitted light for $\theta =$30$^{\circ }$; (b), (c), (d) the wavelength dependencies of the characteristics of the polarization state of the transmitted beam calculated using the ellipsometry data for $\phi =10^{\circ }$. Angles of incidence are indicated on the panels.

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This approach allows one to calculate the electric field of the transmitted light as: $\begin {pmatrix} E_{p}^{out} \\ E_{s}^{out} \end {pmatrix}=T_{ss}\begin {pmatrix} T_1 & T_2 \\ T_3 & 1 \end {pmatrix}\cdot \begin {pmatrix} E_{p}^{in} \\ E_{s}^{in} \end {pmatrix}$. To investigate the HMM birefringence, we used the linearly polarized incident beam with the polarization plane inclined to $\phi =10^{\circ }$ with respect to the plane of incidence, so that both p- and s-components of the incident wave were $E_{p}^{in}=E_0 \cos \phi$, $E_{s}^{in}=E_0 \sin \phi$. It was found that the polarization of the outgoing beam is elliptical. Figure 2(b) shows the spectra of the ratio of the absolute values of the p- and s-polarized components of the transmitted beam, $|E_p^{out}|/|E_{s}^{out}|$, for different angles of incidence. The most pronounced feature here is the minimum near the ENZ point ($\lambda$=810 nm) associated with a strong absorption of the p-polarized radiation within the structure due to the longitudional plasmon resonance excitation. Linear dichroism of nanorods-based HMM is specific for oblique incidence, grows with $\theta$ angle increase and leads to the polarization rotation of the transmitted light.

Modulation of polarization independent on absorption is the phase difference between the outgoing p- and s-polarized components $E_p^{out}$ and $E_{s}^{out}$, $(\delta _p-\delta _s)$; Fig. 2(c) shows the corresponding spectra for normal and oblique incidence. As expected, at normal incidence $(\delta _p-\delta _s)$ coincides with that of the incident wave. For oblique incidence, two jumps of the phaseshift close to the wavelengths of 540 nm and 810 nm appear, which leads to rotation of the polarization ellipse. Using the data shown in Figs. 2(b) and 2(c), the rotation of the polarization ellipse axes was calculated as $\alpha =1/2 \cdot \arctan (\frac {2|E_p^{out}|/|E_{s}^{out}|\cdot \cos (\delta _p-\delta _s)}{(|E_p^{out}|/|E_{s}^{out}|)^2-1})$ (see Fig. 2(d)). Again, there is no polarization plane rotation at normal incidence (Fig. 2(d), black line) and $\alpha \approx \phi =10^{\circ }$ in a wide spectral range. At oblique incidence, the enhancement of $\alpha$ up to 50$^{\circ }$ with respect to the polarization plane of the incident beam is obtained near the ENZ point for $\theta =45^{\circ }$.

3. The revealed HMM features can be exploited for the enhancement of small polarization changes of the incident wave. As a bright example, we studied the enhancement of the Faraday effect in birefringent “HMM$+$Ni film” (Figs. 3(a) and 3(b)) as the p-polarized incident light illuminated first the Ni film, so that s-polarized field component appeared due to the magneto-optical interaction, then the beam passed through the birefringent HMM. The wavelength-angular spectrum of the magnetic contrast shows that a strong enhancement of the magnetic contrast at $\theta =$30$^{\circ }$ and 45$^{\circ }$ at the ENZ point. At the same time, the reference magnetic sample (15 nm thick Ni layer deposited on an Au-free template) demonstrated the Faraday rotation angle of about $0.1^{\circ }$ (Fig. 3(c), black line) in the whole spectral range. We calculated the polarization plane rotation of radiation transmitted through the “HMM$+$Ni film” using the ellipsometry data for $\theta =45^{\circ }$ and the procedure described above, the initial $\phi$ value corresponded to the Faraday effect in the reference Ni film. The polarization ellipse rotation spectrum is shown in Fig. 3(c). Obviously, the Faraday rotation of the composite “HMM$+$Ni” film exceeds that of the Ni film by two orders of magnitude, the strongest enhancement occurs in the vicinity of the ENZ point.

 

Fig. 3. (a) Angular-wavelength spectrum of $\rho$ in Faraday geometry for the p-polarized fundamental beam, the values of the $\rho$ in percents correspond to the colour scale; (b) cross-sections of the angular-wavelength spectrum for $\theta = 0,30$, and $45^{\circ }$; (c) Faraday rotation spectrum of the Ni film (black curve, right axis) and corresponding rotation of the axis of the polarization ellipse $\alpha$ in the “HMM$+$Ni film” (red curve, left axis), $\theta =45^{\circ }$.

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4. Discussion

Firstly it should be emphasized that the studied HMM is an uniaxial anisotropic medium with the optical axis parallel to nanorods, so that at oblique incidence of s- or p-polarized ($\phi =90^{\circ }$ or $\phi$=0$^{\circ }$) radiation either ordinary or extraordinary waves exist and the polarization of the transmitted light remains unchanged. When the incident wave contains s- and p-polarization components, both ordinary and extraordinary waves with different complex propagation constants $q_z^{TE}$ and $q_z^{TM}$ exist and the total field in the HMM is $\vec {E} (x, z)=\vec {E}^{TE} (x, z)+\vec {E}^{TM} (x, z) =\left (\vec {A}^{TE}exp(iq_z^{TE}z)+\vec {A}^{TM}exp(iq_z^{TM}z)\right ) exp(ik_x x).$ As a result, giant changes of the light polarization appear due to the two mechanisms that involve (1) different phase velocities of the TE and TM modes and (2) anisotropy of the light absorption, specific to arrays of plasmonic nanorods. In the first mechanism, the HMM acts as an ordinary birefringent crystal with the largest variations of $\Delta q_z=Re(q_z^{TE}-q_z^{TM})$ near the ENZ point. It leads to the appearance of the jumps in the accumulated phase difference of the TE and TM waves $(\delta _p-\delta _s)(\lambda )=L \Delta q_z$ (Fig. 2(c)). The second mechanism involves stronger absorption of the p-polarized light as compared to the s-polarized one. Importantly that both processes affect the polarization state of the transmitted light simultaneously: the linear dichroism governs the polarization plane rotation, whereas $\Delta q_z$ provides the polarization ellipticity. Thus the HMM can work as an efficient converter from linear to elliptically polarized light close to the ENZ spectral point.

These mechanisms are confirmed by the simulations of the TE/TM modes’ propagation constants. Figure 4 shows the angular-wavelength spectra of the $Re(q_z^{TM}, q_z^{TE})$, calculated from the Maxwell equations within the effective medium model. For the TE (ordinary) mode, the parameter $Re(q_z^{TE})$ has no peculiarities in the vicinity of the ENZ and has a maximum at the ENP point, which is consistent with the $\epsilon _{\perp }$ spectra shown in Fig. 1(c). On the contrary, for the TM (extraordinary) mode a strong modulation of $Re(q_z^{TM})$ takes place in the vicinity of the ENZ point due to the sign reversal of $\epsilon _{\parallel }$. The difference $\vert Re(\Delta q_z)\vert$ (the color map in Fig. 4(a)) vanishes at $\theta = 0^{\circ }$ and grows rapidly for higher angles of incidence, thus increasing the HMM birefringence. This leads to the leaps in the $\delta _p-\delta _s$ spectra that were observed experimentally and modulation of the polarization ellipse rotation (Figs. 2(c) and 2(d)). As well large difference of $Im(q_z)$ for TM and TE modes manifests enhanced linear dichroism (Fig. 4(b)). It should be noted that linear dichroism can occur in nonhyperbolic elongated plasmonic structures, whereas such extremely high values of $Re(\Delta q_z)$ should be attributed only in hyperbolic strongly anisotropic media in the spectral vicinity of the ENZ point.

 

Fig. 4. Calculated angular-wavelength spectra of the (a) $Re(q_z)/(\omega /c)$ and (b) $Im(q_z)/(\omega /c)$ values for TE and TM modes in HMM; bottom map in (a) is $Re(\Delta q_z)/(\omega /c)$.

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Additional possibility for the control over the light polarization in the“HMM$+$Ni film”, was realized even if the structure is irradiated by purely p-polarized light from the Ni side. Application of the magnetic field in the Faraday geometry leads to the rotation of the polarization plane of light passed through the magnetic film, followed by the birefringent HMM that screws it up. The Faraday rotation in 15 nm thick Ni film does not exceed 0.1$^{\circ }$, whereas it turns out to be enough to provide the polarization plane rotation of the “HMM$+$Ni film” sample up to several degrees (Fig. 3(c)). This is the manifestation of amplification of the Faraday rotation by a birefringent HMM enhanced close to the ENZ point. It is worth noting that the polarization changes are maximal at the ENZ point, and decrease when moving towards the hyperbolic dispersion region. We emphasize that the real and imaginary parts of $q_z^{TM}-q_z^{TE}$ are responsible for each feature separately, i.e. $\rho$ is enhanced by the HMM linear dichroism, whereas large difference of the phase velocities for the TE and TM light waves propagating in the HMM modulates the sign of the magnetic contrast. High ellipticity of the transmitted light doesn’t allow us to evaluate the polarization plane rotation angle from the magnetic contrast. Spectra of the magnetic contrast (Figs. 3(a) and 3(b)) and polarization ellipse rotation in nonmagnetic sample (Fig. 2(d)) are qualitatively similar the $\rho$ enhancement close to ENZ can be used for the magnetic-field control over the light intensity passing through the HMM.

5. Conclusion

Summing up, we studied the polarization effects in HMM composed of arrays of Au nanorods in porous alumina templates and that supplemented by a continuous ferromagnetic Ni film. Specific spectral features associated with the hyperbolic dispersion of these structures bring about two spectral resonances that correspond to epsilon-near-pole (surface plasmons excitation) at 540 nm and epsilon-near-zero spectral point close to 810 nm, so that the hyperbolic dispersion regime is achieved at longer wavelengths. Manyfold enhancement of the magneto-optical Faraday effect in “HMM$+$Ni film” was observed in the spectral vicinity of the ENZ point, thus providing perspectives for applications of HMM for the control over the polarization state of light.