1. Introduction

Recently, the epsilon-near-zero (ENZ) metamaterials, i.e., metamaterials with near-zero permittivity, emerge into the focus of the extensive exploration because of the anomalous electromagnetic features in microwave and optical frequencies. For instance, the ENZ metamaterials can squeeze and tunnel the electromagnetic energy through narrow waveguide channels [1–4] and shape the radiation wavefront [5] due to the near-zero phase variation of the electromagnetic wave inside the ENZ metamaterials. Particularly, the ENZ metamaterials can be applied in the invisible cloaking [6, 7], the design of displacement current insulation [8], and the sub-wavelength image transporting [9]. Besides, the ENZ metamaterials can also be used to significantly enhance the optical nonlinearities [10] and the photonic density of states [11]. In theory, the electromagnetic properties of the ENZ metamaterials are generally described by the effective medium theory (EMT) [12] for the meta-atom structures of the metamaterials being much smaller than the wavelength. However, the variation of the electromagnetic field on the scale of the meta-atom structure will in fact result in the optical nonlocality with strong spatial dispersion, thus the permittivity components will depend on both frequencies and wave vectors [13]. Recently, the optical nonlocalities in nanowire metamaterials [14] and metal-dielectric multilayer metamaterials [15–20] around the ENZ frequency have been investigated.

In this work, the optical nonlocality is theoretically and experimentally studied in the metal-dielectric multilayer metamaterials with respect to the TE-polarized incident light of different angles of incidence. The analytical description about the optical nonlocality is developed based on the transfer-matrix method. The physical mechanism of the difference between the nonlocal effective permittivity and the EMT-based effective permittivity depending on the incident angle is revealed based on the analysis of the band structure, the dispersion relation, and the iso-frequency contours (IFCs) of the metal-dielectric multilayer metamaterials. Furthermore, such effective permittivity difference as a function of the angle of incidence is also retrieved in the metal-dielectric multilayer metamaterials according to the measured transmission and reflection spectra, which agrees with the theoretical prediction.

2. Fabrication of metal-dielectric multilayer stack

Figure 1(a) illustrates the schematic of a 4-pair periodic metal-dielectric multilayer stack, composed by gold (Au) and alumina (Al₂O₃), with respect to the TE-polarized incident light. The permittivity and the thickness of the Au layer and of the Al₂O₃ layer are denoted as (ε₁, d₁) and (ε₂, d₂), respectively. The thickness of the Au layer and of the Al₂O₃ layer is individually designed to be 15 nm and 70 nm. The multilayer stack is deposited on top of quartz (SiO₂) substrates with the electron-beam evaporation system, where Au is deposited at the rate of 0.4 Å/sec and Al₂O₃ is deposited at 0.2 Å/sec. Each material is individually deposited on a quartz substrate first to calibrate and optimize the deposition parameters for the electron-beam evaporation system (Kurt J. Lesker). The optical constant of each material and the film thickness are characterized with the variable angle spectroscopic ellipsometry (VASE, J. A. Woollam Co. VB400/HS-190). The VASE measurements show that the optical constant of Au matches the standard data of Johnson and Christy [21] based on the fitting from a general oscillator model. The dielectric constant of Al₂O₃ is fitted from the Cauchy dispersion relation. The VASE measured film thickness for each material also matches the thickness value for the set deposition parameters. Figure 1(b) shows the scanning electron microscope (SEM) picture of the cross section of the fabricated Au-Al₂O₃ multilayer stack, in which the focused ion beam (FIB) system (Helios Nanolab 600) is used to cut the cross section. Each deposited thin layer can be clearly seen, where the bright and the dark stripes individually correspond to the Au layers and the Al₂O₃ layers. The thickness of the deposited layers can be characterized with the VASE, and the measured averaged thickness for the Au layer and the Al₂O₃ layer is 15 ± 0.4 nm and 70 ± 1.6 nm, respectively.

 

Fig. 1 (a) Schematic of the Au-Al₂O₃ multilayer stack on a quartz substrate. (b) The SEM picture of the cross section of the fabricated 4-pair Au-Al₂O₃ multilayer stack.

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3. Theoretical analysis and experimental characterization

In the long wavelength limit, the electromagnetic properties of the metal-dielectric multilayer stack can be simply described as an effective medium following a simple dispersion relation as

According to Eq. (3), Fig. 2 displays the band structure [Figs. 2(a) and 2(b)] and the dispersion relation [Fig. 2(c)], together with the transmission [Fig. 2(d)], reflection [Fig. 2(e)] and absorption [Fig. 2(f)] spectra of the Au-Al₂O₃ multilayer stack with the optical loss of Au for different angles of incidence θ = 0° and θ = 50°. It is noted that the frequency and the wave vector are normalized by the plasma frequency of Au ω_p = 1.3666 × 10¹⁶rad/s and the corresponding plasma wave vector k_p = ω_p/c. The light blue region represents the wavelength range from 450 nm to 750 nm in the measurements. Figure 2(a) displays the real part of the band structure of the multilayer stack in the first Brillouin Zone (BZ). In general, the peaks in the transmission spectrum should be located at the frequencies that satisfy the Bragg’s law, i.e., the points located at the center and the boundary of the first BZ. Meanwhile, the imaginary part of the band structure [Fig. 2(b)] implies the propagation loss in the multilayer stack, which is inversely proportional to the magnitude of the transmission peaks. Therefore, there is only one transmission peak for the Au-Al₂O₃ multilayer stack, corresponding to the frequency (marked by arrows) that satisfies the Bragg’s law with the low propagation loss. With respect to the variation of the angle of incidence, the optical path of each layer in the multilayer stack varies, leading to the shift of the band structure to a higher frequency for both the real part and the imaginary part, which causes the up shift of the transmission peak at the frequency from the red arrow to the purple arrow. Importantly, the frequency where $\text{Re}\left({\epsilon }_{\text{eff}}^{\text{nonloc}}\right)={\text{sin}}^2\theta$ and Re(k_z/k_p) = Im(k_z/k₀) also varies corresponding to the change of the band structure with respect to the angle of incidence, as marked by the dots. In the dispersion relation of Fig. 2(c), the variations of all frequencies that satisfy the Bragg’s law in the real part of the band structure are plotted as a function of the wave vector k_x (= k₀ sinθ), with the first branch showing the frequency where $\text{Re}\left({\epsilon }_{\text{eff}}^{\text{nonloc}}\right)={\text{sin}}^2\theta$. Since the light lines represent different angles of incidence, the variations of the frequency where $\text{Re}\left({\epsilon }_{\text{eff}}^{\text{nonloc}}\right)={\text{sin}}^2\theta$ and the transmission peak can be straightforwardly represented as the intersection points between the dispersion curves and the light lines at different angles of incidence, as displayed in the first branch marked with the dots and the second branch marked by the arrows, respectively. The transmission, reflection and absorption spectra of the Au-Al₂O₃ multilayer stack are then measured in the wavelength range from 450 nm to 750 nm, with respect to different angles of incidence from 0° to 80° with a variation of 10°. It is shown in Fig. 3 that the experimental measurements (black-solid curves) match the theoretical predictions (red-dashed curves), including the shift of the transmission peak with respect to the variation of the angle of incidence.

 

Fig. 2 (a) The real part and (b) the imaginary part of the band structure of the Au-Al₂O₃ multilayer stack. (c) The dispersion relation of the Au-Al₂O₃ multilayer stack. (d) The transmission, (e) reflection, and (f) absorption spectra of the Au-Al₂O₃ multilayer stack.

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Fig. 3 (a) The transmission, (b) reflection, and (c) absorption spectra of the Au-Al₂O₃ multilayer stack in experiments (black curves) and in theory (red curves) at different angles of incidence from 10° to 80°.

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In order to illustrate the optical nonlocality, Fig. 4(a) plots the difference between the nonlocal ENZ wavelength ${\lambda }_{\text{ENZ}}^{\text{nonloc}}$ calculated from Eq. (4) and the EMT-based ENZ wavelength ${\lambda }_{\text{ENZ}}^{\text{EMT}}$ according to Eq. (2) as a function of the angle of incidence. Compared to the constant EMT-based ENZ wavelength ${\lambda }_{\text{ENZ}}^{\text{EMT}}=634.241\hspace{0.17em}\text{nm}$, the difference of nonlocal ENZ wavelength varies from 22.471 nm to 21.178 nm as the angle of incidence increases from 0° to 80°. It is shown that the nonlocal ENZ wavelength of the metal-dielectric multilayer stack depends on not only the actual layer thickness but also the angle of incidence for the TE-polarized incident light. On the other hand, Fig. 4(b) displays the difference of the effective permittivity defined in Eq. (5) as a function of the angle of incidence, at a fixed wavelength of ${\lambda }_{\text{ENZ}}^{\text{nonloc}}(\theta =0°)=656.712\hspace{0.17em}\text{nm}$. Clearly, with the increase of the angle of incidence, the difference of the effective permittivity varies gradually from 0.278 to 0.263 due to the optical nonlocality.

 

Fig. 4 (a) The difference between the nonlocal ENZ wavelength and the EMT-based ENZ wavelength as a function of the angle of incidence. (b) The real part of the effective permittivity difference at the nonlocal ENZ wavelength of θ = 0° according to Eq. (5) with respect to different angles of incidence. (c) The real part of the measured effective permittivity difference (black triangles) compared to the theoretical prediction (red circles) at different angles of incidence. The IFCs at different wavelengths where $\text{Re}\left({\epsilon }_{\text{eff}}^{\text{nonloc}}\right)={\text{sin}}^2\theta$ and Re(k_z/k_p) = Im(k_z/k₀) at the incident angles of (d) θ = 0° and (e) θ = 50°.

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According to the approximated approach based on incoherent interference of optical beam oscillated inside a slab, the effective propagating refractive index of the Au-Al₂O₃ multilayer stack ñ = n + iκ (with ñ = k_z/k₀) can be retrieved from the measured transmission and reflection spectra as [24]

The variation of the wavelength where Re(k_z/k_p) = Im(k_z/k₀) with respect to the angle of incidence can be explained according to the analysis of the IFCs based on Eq. (3). Here two IFCs with respect to the angle of incidence θ = 0° and θ = 50° are given in Figs. 4(d) and 4(e), where the light cones are denoted as green circles and the real parts and the imaginary parts of the wave vector k_z are indicated as red curves and black curves, respectively. The point where Re(k_z/k_p) = Im(k_z/k₀) is marked with red dot in the IFCs. As the angle of incidence increases from θ = 0° to θ = 50°, the wave vector k_x varies from 0 to k₀ sinθ. According to the momentum conservation, the corresponding wave vector k_z varies from 0.0620227(1 + i)k_p to 0.0808913(1 + i)k_p and the wavelength shifts from 656.712 nm to 603.485 nm.

4. Conclusions

The optical nonlocality in the metal-dielectric multilayer metamaterials for the TE-polarized incident light is theoretically studied based on the transfer-matrix method, with the analysis of the band structure, the dispersion relation, and the IFCs. The measured transmission and reflection spectra with respect to different angles of incidence for the fabricated Au-Al₂O₃ multilayer stack agree with the theoretical analysis. The difference between the nonlocal effective permittivity and the EMT-based effective permittivity depending on the angle of incidence is observed according to the retrieved refractive index results from the measured transmission and reflection spectra, which directly demonstrates the optical nonlocality in the metal-dielectric multilayer stacks.