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Abstract
Hyperbolic metamaterials (HMMs) show great promise in photonics applications because their unconventional open isofrequency surface permits enlargement of wavenumbers without limitation. Although optical behaviors in HMMs can be macroscopically described by theoretical calculations with the effective medium approximation (EMA), neglect of microscopic phenomena in each layer leads to discrepancies from exact numerical results. We clarify the origin of bulk propagating waves in HMMs and we show that they can be classified into two modes: long- and short-range surface-plasmon-based coupled modes (LRSP and SRSP, respectively). Especially, we find that the ratio of the number of LRSP and SRSP couplings dominates the property of each propagation mode. This plasmonic interpretation bridges the gap between the EMA and numerical exact solutions, thereby facilitating studies on HMM applications.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Enlargement of wavenumbers is a key area of research on nanophotonics, because it results in strong field confinement and enhancement, as well as an increment in the photonic density of states [1–6]. Generally, dielectric materials have a closed isofrequency surface, such as a sphere or spheroid, leading to a restriction on the increase in wavenumbers. Plasmonic materials on the other hand, permit wavenumbers to increase to infinity by taking one or two components of wavenumber imaginary numbers. Consequently, isofrequency surfaces in plasmonics are open, but at least one axis is an imaginary axis [7–9].
Hyperbolic metamaterials (HMMs) have attracted fundamental interest, with the expectation that they might open a door to a new phase of photonics research. This is because of their unconventional isofrequency surfaces, which permit the enlargement of wavenumbers without limit under the assumption of lossless case. Even though we take material loss into consideration, much larger wavenumbers than conventional materials can be obtained. HMMs have a strong anisotropy, where one of the principal components of the permittivity tensor has the opposite sign to the other two {e.g., [ε] = diag(ε//, ε⊥, ε⊥) with ε//ε⊥□< 0}, leading to an open isofrequency surface [10–13]. HMMs can be realized by several structural designs, for example, alternative metal–dielectric multilayered structures, plasmonic nanorod arrayed structures, or natural hyperbolic materials [14–18].
For theoretical calculations on metamaterials as effective media, the effective medium approximation (EMA) is generally used to render such calculations simple and intuitive [19–22]. Although the EMA qualitatively expresses the properties of HMMs, it sometimes shows a quantitative deviation from the exact solution due to its neglect of microscopic plasmonic phenomena. To investigate optical propagation modes in HMMs, it is necessary to consider not only the macroscopic envelope field, but also microscopic plasmonic couplings on each interface. In addition, although Bloch’s condition for periodicity is a useful construct that includes boundary conditions on each interface, it assumes the structure is a bulk [23–29]. Consequently, this approach is unsuitable for investigating nanoscale structures.
Propagation modes supported by alternative metal–dielectric multilayered structures can be studied by means of numerical calculations [30–32]. Although numerical calculations provide sufficiently exact solutions, there is also a need to explain these solutions theoretically. Furthermore, although studies on the limitations of the EMA have been reported, we stress that these do not adequately explain differences between the exact solutions and solutions produced by using the EMA [27–29]. An examination of the plasmonic couplings of propagation modes should reveal the cause of differences between the exact solutions and those produced with the EMA. It should also permit us to provide a theoretical intuitive explanation of numerical calculations. Elucidation of the optical behavior of HMMs by bridging the gap between macroscopic and microscopic interpretations has not previously been reported, although it is recognized this will be necessary to promote further studies on HMM-based devices and their applications.
Here, we study propagation modes in multilayered-type HMMs. First, we numerically calculate the properties of the propagation modes in detail and we compare these results to those calculated by using the EMA. We then discuss two types of propagation mode, long-range surface plasmon (LRSP)-based modes and short-range surface plasmon (SRSP)-based modes, to clarify the cause of differences between the EMA-based solutions and the exact solutions. Especially, we find the ratio of the number of LRSP and SRSP couplings is a crucial element to decide the properties of each propagation mode. We also consider bulk HMMs, to confirm that the exact solutions are consistent with theoretically calculated Bloch waves.
2. Propagation modes by EMA
Figure 1(a) and 1(b) are schematic and cross-sectional views, respectively, of a planar optical waveguide with a HMM core and a cladding of air. The HMM core consists of alternating multilayers of metal (thickness dm) and a dielectric material (thickness dd), with a total thickness of h. The direction of propagation of optical waves along the core is shown by the red arrow.
Fig. 1 Schematic view of the structure studied and its effective permittivity. (a) Schematic representation of a planar optical waveguide with a HMM core. The direction of propagation of light is shown by the red arrow. (b) A cross-section of (a). (c) Derived real parts of the permittivity tensor components of the multilayered structure as a function of the metal filling ratio f. The structure consisted of Au and SiO2, and had an operating wavelength λ0 = 1550 nm. Inset: Schematic isofrequency surfaces of this multilayer structure in wavenumber space.
By using the EMA, the components of the permittivity tensor of multilayered structures [ε] = diag(ε//, ε⊥, ε⊥) can be written as follows:
where ε// and ε⊥ are the pertmittivity components parallel and perpendicular to the anisotropy axis (x-axis), resoectively. Also, εm and εd are the permittivities of the metal and dielectric media in the multilayered structures, respectively. f = dm/(dd + dm) is the metal filling ratio [19–21].Figure 1(c) shows the derived real part of ε// and ε⊥ of a multilayered structure consisting of Au (εm = –114.5 + 11.01i) and SiO2 (εd = 1.4442) as a function of f at the vacuum wavelength λ0 = 1550 nm [33,34]. Because the permittivity tensor of the multilayered structure maintains ε// > 0 and ε⊥ < 0 throughout f, the structure can be recognized as a Type II HMM with an opened isofrequency surface in wavenumber space, as shown in the inset to Fig. 1(c). The isofrequency surface is derived by means of a dispersion relation written as kx2/ε⊥ + (ky2 + kz2)/ε// = ω2/c2. Hence, in this study, we performed all our calculations with f = 0.5 at λ0 = 1550 nm.
Next, we calculate propagation modes in the HMM-core optical waveguide, assuming sinusoidal optical waves propagating along the z-direction. By using the EMA, the Helmholtz equations inside core and cladding are derived as follows:
where β, ω, c, and ε0 are the propagation constant, the angular frequency, the velocity of light in a vacuum, and the permittivity of the cladding, respectively. Supposing propagating wave and evanescent wave inside core and cladding, respectively, we can obtain the general solutions from Eq. (2). By imposing the boundary conditions, the characteristic equation of the HMM-core optical waveguide is derived as follows:Figure 2(a) shows the effective index neff ( = β/k0) as a function of the normalized core thickness h/λ0 derived from Eq. (3) at λ0 = 1550 nm. The first and second parts of left-hand side of the characteristic equation in Eq. (3) show even and odd order modes, respectively. Here, we show only the first two orders of modes, whereas HMM-core optical waveguides support many propagation modes. The electric (Ex) and magnetic (Hy) field distributions of each mode at h = 165 nm (h/λ0 ≈0.1) are shown in Fig. 2(b).
Fig. 2 Propagation modes calculated by the EMA at λ0 = 1550 nm. (a) neff of the zero- and first-order modes as a function of the normalized HMM thickness, calculated by using the EMA. Inset: the cross-section of an HMM-core optical waveguide. (b) The electric Ex and magnetic Hy field distributions of the zero- and first-order modes at h = 165 nm and f = 0.5.
We can show that neff increases to infinity as the core thickness decreases. Note that all the orders of the modes show the same trend as the zero- and first-order modes shown in Fig. 2(a). As can be seen in Fig. 2(b), Ex and Hy field distributions of both the zero- and first-order modes exhibit sinusoidal profiles inside the core and an evanescent wave in the cladding. Such field profiles are called ‘bulk propagating waves’, and are completely different from plasmonic surface waves.
3. Exact propagation modes by the finite-element method
3.1 Difference between EMA and exact solutions
To confirm the accuracy of the propagation modes calculated by the EMA, we derived exact solutions for the propagation modes in a HMM-core optical waveguides with a period of 30 nm by means of finite-element method (FEM) calculations, performed by using commercially available software: FemSIM (Synopsys Inc., Pasadena, CA, USA) [35,36].
Figure 3 shows neff, as calculated by the FEM, as a function of the core thickness. To change the core thickness, we varied the number of metal layers Nm with a fixed period. As can be seen in Fig. 3, we identified two types of propagation mode: lower- and higher-index modes that show opposite dependencies on the core thickness. The higher-index modes show completely different behaviors from the results calculated by the EMA [Fig. 2(a)]. On the other hand, the lower-index modes show a similar trend to that in Fig. 2(a), but the index values differ from those obtained by the EMA [Fig. 2(a)]. We also performed the same calculations as those in Fig. 3 but with changes in the period (not shown). In the case of isofrequency surfaces, the exact solutions approach the value given by the EMA as the period decreases [27–29]. However, in our calculations regarding neff, the exact value of lower-index modes does not approach to the value obtained by the EMA [Fig. 2(a)], even though the period is decreasing. We therefore conclude that the EMA does not take the exact values of neff in either the lower- or the higher-index mode.
Fig. 3 neff vs. HMM core thickness calculated by the FEM for f = 0.5 at λ0 = 1550 nm. The core thickness was varied by changing Nm with a constant period of 30 nm.
Ex and Hy field distributions are shown in Fig. 4 for both lower- and higher-index modes at h = 165 nm and Nm = 17 with a period of 10 nm. Although they have sinusoidal envelope fields, such as that shown in Fig. 2(b), they contain short period oscillatory profiles inside the envelopes; the short period oscillatory profiles of higher-index modes differ from those of the lower-index modes.
Fig. 4 The electric and magnetic field distributions of propagation modes in a planar HMM, calculated by the FEM at λ0 = 1550 nm. (a) Ex and (b) Hy field distributions of the lower-index modes. (c) Ex and (d) Hy field distributions of the higher-index modes. These are the waveguide cross-section. The total thickness of the core h = 165 nm, which is the subwavelength size (h/λ0 ≈0.1) at a period of 10 nm at f = 0.5.
From the above results pertaining to Figs. 3 and 4, we can conclude that the short period oscillatory profiles inside the envelopes are the cause of the discrepancies between the EMA-based and exact solutions.
3.2 Microscopic interpretation of propagation modes
As shown in Fig. 4, the field distributions of the lower- and higher-index modes differ from one another. We attribute this to microscopic behaviors of the coupled surface plasmon polaritons (SPPs). In considering the coupled modes of SPPs, we start from a metal film (dielectric–metal–dielectric: DMD) waveguide and a metal gap (metal–dielectric–metal: MDM) waveguide, in which the plasmonic couplings are either symmetric or asymmetric. With regard to a DMD plasmonic coupling (for an intermediate Au layer), both symmetric and asymmetric coupling modes (i.e., LRSP and SRSP) can exist. On the other hand, in the case of MDM plasmonic coupling (with an intermediate SiO2 layer), only a gap surface plasmon (GSP) can exist, because the other coupling mode has a cutoff in a subwavelength-sized layer. Hy field profiles for LRSP, SRSP, and GSP modes are shown in Fig. 5(a).
Fig. 5 Schematic illustration of the magnetic field profiles Hy for various plasmonic coupled modes. (a) LRSP, SRSP, and GSP. (b) LG mode and SG mode. The white area is dielectric and the gray area is metal.
Multilayered-type HMMs can be considered as aggregated structures of MDM and DMD units. We therefore need to consider new types of coupled modes between DMD and MDM plasmonic couplings: the LRSP–GSP mode (named the LG mode) and the SRSP–GSP mode (named the SG mode). Hy field profiles of these modes are shown schematically in Fig. 5(b).
By comparing Fig. 4 with Fig. 5, we find that the lower-index modes show the same field distribution as the LG modes. On the other hand, the higher-index modes show the same field distribution as the SG modes. Hence, the lower- and higher-index modes in Fig. 3 correspond to the LG and SG modes, respectively. In addition, we introduce the mode order p as a subscript, in LGp and SGp; e.g., LG0, LG1, SG0, and SG1. These microscopic plasmonic couplings are the origin of the differences between solutions calculated by the EMA [Fig. 2(a)] and the exact solutions calculated by the FEM [Fig. 3].
4. Origin of the LG and SG modes
4.1 Transition process to LG and SG modes
To investigate the transition process of LG and SG modes in detail, we analyzed the propagation modes in alternating metal–dielectric multilayers by the method reported by Avrutsky and associates [30]. This system consists of multilayers with a gradually changing thickness of the SiO2 layers and increasing Nm, as shown in Fig. 6(a). Figure 6(b) shows the neff of propagation modes as a function of the SiO2 layer thickness d and Nm, where the Au layer thickness is fixed at 15 nm.
Fig. 6 The transition and generation process of propagation modes in a planar HMM. (a) Schematic illustration of this calculation system of decreasing the SiO2 layer thickness d and increasing Nm. (b) The transition of neff of propagation modes at λ0 = 1550 nm. The thickness of the Au layers is 15 nm and d is varied from 2 μm to 15 nm. neff of LRSP-like and SRSP-like modes gradually approach the same value, whereas the neff of other modes (r, s) increase with increasing Nm.
At Nm = 2, three modes exist: an LRSP-like mode, an SRSP-like mode, and an SG-like mode, as shown in Fig. 7. The LRSP-like and SRSP-like modes are not simple DMD coupling modes as shown in Fig. 5(a), as their total field profiles and characteristics are dominated by LRSP and SRSP on a single metal film. We call the other mode the ‘SG-like mode’ because it possesses a similar field profile to the SG mode shown in Fig. 5(b), but we note that the cladding and dielectric layers are not the same materials in this structure.
Fig. 7 Hy field profile of LRSP-like, SRSP-like, and SG-unit-cell-like mode (0, 0) at Nm = 2 and d = 150 nm
At Nm = 3, a new mode is generated with one LRSP coupling, whereas the SG-like mode increases one SRSP coupling. The mode generation continues on increasing Nm to 4 or more in the same way: as each additional Au layer is added, a new mode occurs with LRSP couplings, whereas the existing modes obtain one SRSP coupling. In the meantime, we can realize that all modes except the LRSP-like and SRSP-like modes have a GSP coupling at every intermediate SiO2 layer and an SRSP couplings at both end layers. Hence, to distinguish each mode, we use the representation (l, s) where l and s indicate the number of LRSP and SRSP couplings, respectively, on the individual Au internal layers (SRSP couplings at the two end layers are excepted).
Consequently, lower neff modes exhibit properties originating from LRSP, whereas higher neff modes show properties originating from SRSP. This is the origin of the LG and SG modes. As shown in Fig. 6(b), we observe that LG and SG modes can be classified by two indices, l and s. In addition, the number of LRSP couplings l and SRSP couplings s represent the numbers of nodes in higher (SG)- and lower (LG)-index modes, respectively. Hence, (l, 0) and (0, s) correspond to LG0 and SG0, respectively. Similarly, (l, 1) and (1, s) correspond to LG1 and SG1, respectively. Although LG and SG modes have different characteristics, such cascaded transition process suggests that they are substantially identical modes, and their difference can be attributed to the ratio of the number of LRSP and SRSP couplings.
4.2 Structural dispersion of LG and SG modes
So far, we have discussed the origin of bulk propagating waves in HMMs by considering only the field distributions. Such microscopic interpretations of the LG and SG modes can be corroborated by comparing their neff values to LRSP, SRSP, and GSP. We therefore also calculated the structural dispersions with the FEM by varying the thickness of the Au and SiO2 layers (i.e., varying the period) for a fixed Nm. Figure 8 shows the neff for the zero- and first-order of the LG modes [Fig. 8(a)] and the SG modes [Fig. 8(b)] as a function of the period for the condition Nm = 11. For comparison, neff for GSP, SRSP, and LRSP are also plotted in Fig. 8.
Fig. 8 neff vs. period at λ0 = 1550 nm calculated by the FEM. (a) neff of LG0 and LG1 with respect to the period at f = 0.5. (b) neff of SG0 and SG1 with respect to the period with fixed Nm = 11 and f = 0.5. neff of GSP (orange dashed–dotted line), SRSP (green dashed line), and LRSP (purple dotted line) are also plotted as functions of the SiO2 gap and the Au film thickness, which are half the size of a period.
The dispersion curves of the LG and SG modes can be quantitatively explained in terms of GSP, LRSP, and SRSP, because these modes are basic components. In the case of GSP, it is known that neff increases with decreasing gap distance [37,38]. In the case of LRSP and SRSP, it is known that neff decreases slightly or increases with decreasing thickness of the metal layer [39–41]. Because the LG0 mode consists of GSP and LRSP couplings, the neff line of LG0 is seen between the lines for GSP and LRSP in Fig. 8(a). By comparing LG0 and LG1, the fact that LG1 has a higher neff than LG0 can be attributed to one SRSP coupling in LG1, which pushes up the line to a higher value, where LG1 is the (1, s) mode in Fig. 6(b). Similarly, as can be seen in Fig. 8(b), whereas the DMD couplings of SG0 are all SRSP couplings, SG1 has one LRSP coupling, which pushes down the line to a lower value, where SG1 is the (l, 1) mode in Fig. 6(b). By comparing the behaviors of neff to LRSP, SRSP, and GSP, we can reinforce the microscopic interpretations of LG and SG modes beyond a consideration of the field distributions.
5. Comparison to Bloch waves
Finally, we discuss a waveguide with an infinite core thickness, i.e. a bulk HMM, as shown in Fig. 9(a). In such waveguides, propagating waves as SPPs are usually called Bloch waves. The characteristic equation of propagation modes in bulk HMMs are derived from Maxwell’s equations by applying boundary conditions at each layer and Bloch’s condition for periodicity, as follows:
where kB is the Bloch wavenumber. If we assume that kB = 0 or kB = π/(dd + dm), the dispersion relation of Eq. (4) is reduced to the following expression:From Eq. (5), we can obtain the relationship between the period dd + dm and neff ( = β/k0) of propagation modes in the case of kB = 0 or kB = π/(dd + dm).Fig. 9 Calculated results for Bloch waves in a bulk HMMs. (a) Schematic illustration of a bulk HMM. (b) neff of LG0 and SG0 modes at Nm = 101 and Bloch waves in bulk HMMs with respect to the period at λ0 = 1550 nm. The solid lines are analytically calculated Bloch waves kB = 0 (green line) and kB = π/(dd + dm) (orange line) in bulk HMMs. Diamonds and circles are the neff of the LG0 and SG0 modes calculated by the FEM, respectively.
Figure 9(b) shows neff for a Bloch wave with respect to the unit-cell size in the case kB = 0 or kB = π/(dd + dm). Here, we compare the analytical results with neff for the LG0 and SG0 modes calculated by FEM at a fixed Nm = 101. In FEM calculations, neff approaches a constant value asymptotically with increasing the period. This means that the structure is thick enough to be regarded as a bulk material.
We observe that the LG0 and SG0 modes are in a good agreement with the Bloch wave at kB = 0 and kB = π/(dd + dm), respectively. Such agreements can be explained by a physical picture described below; the Hy field profiles in Figs. 4(b) and 4(d) give us an insight into these consistencies. The Hy field profile of LG0 in a bulk HMM can be recognized as constant at kB = 0, whereas the Hy field of SG0 has nodes at every Au layer with a wavelength of one period at kB = π/(dd + dm). We observe a slight difference between LG0 and the Bloch wave at kB = 0 in Fig. 9(b), especially for smaller periods. The situation of kB = 0 can be realized only when the material is much larger along x-axis than wavelength. The total thickness in smaller period range of FEM calculation is not enough to permit the multilayers to be treated as a bulk material, leading to the slight difference between LG0 and Bloch wave. In contrast, the value of kB = π/(dd + dm) depends only on the period and is independent of Nm.
The solutions for Bloch waves analytically derived from Eq. (5) are in good agreement with the exact solutions obtained by the FEM. This is because Bloch waves include microscopic phenomena at each layer through boundary conditions. We stress that such an agreement suggests the existence of SG and LG modes.
6. Conclusion
In conclusion, we studied the propagation modes supported by multilayer-type HMMs and we found that they can be classified into two different types of coupled plasmonic modes: LRSP-GSP (LG mode) and SRSP-GSP (SG mode). We have investigated their transition process by gradually changing the thickness of the dielectric layer and increasing the number of metal layers. This clarified the microscopic constituents of the LG and SG modes and enabled us to achieve a better understanding of the behaviors of those modes: the ratio of the number of LRSP and SRSP couplings dominates the property of each propagation modes. Lastly, we successfully showed a good agreement between Bloch waves and LG0 and SG0 in a bulk HMM. This enables us to provide a theoretical interpretation of numerically calculated exact solutions and to achieve an intuitive understanding the behavior of light confined in HMMs nanostructures.
We emphasize that although EMA provides a qualitative interpretation of the properties of HMMs, it produces quantitative differences from exact solutions due to its neglect of microscopic plasmonic coupling effects. We believe that further investigations and deep insight into plasmonic propagation modes in HMMs beyond the EMA will promote studies on HMM-based devices and their applications.
Funding
Photonics Advanced Research Center (MEXT); Osaka University.
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