Nonreciprocity and one-way topological transitions in hyperbolic metamaterials

Optical metamaterials are artificial media with engineered electromagnetic response that can be realized through subwavelength structuring, and they facilitate properties normally limited or not found in naturally occurring materials.^1–6 One such property is optical nonreciprocity—a rare and generally weak characteristic of light to differentiate between opposite propagation directions.^7,8 This property, which exists in magnetic materials such as ferrites, is of immense importance for devices such as optical isolators and circulators, which are widely used to stabilize laser operations and to route signals in optical telecommunication networks.^9–11 Therefore, nonreciprocal optical components are of significant interest for optical integration, which can be achieved by combining magneto-optical (MO) materials with resonant nanophotonic^12–17 and plasmonic^18–26 structures. Such integration also allows enhancement of generally weak MO response of ferrites through strong light-matter interactions in photonic^12–17 and plasmonic^18–26 nanostructures and metamaterials.²⁷ 

Metamaterials with magnetic constituents have been used to achieve negative index of refraction and tunable response in an external magnetic field.^28–33 However, very little is known on the nonreciprocal effects that can be engineered using magnetic metamaterials.²⁷ In this context, hyperbolic metamaterials (HMMs),^31,34–44 a class of metamaterials with hyperbolic isofrequency contours, known for their ability to enhance light-matter interaction over a broad spectral range, including broadband asymmetric⁴⁵ but reciprocal⁸ scattering, can be exceptional candidates for novel devices, offering both enhanced nonreciprocal effects and broadband operation which are unattainable with other classes of metamaterials. In this letter, we demonstrate that HMMs with MO activity exhibit unprecedented nonreciprocal characteristics such as one-way topological transitions⁴³ and one-way hyperbolic dispersion regimes.

Optical nonreciprocity occurs only when an optical system lacks both time-reversal symmetry and inversion symmetry.^13,15,16 In the particular case of layered media magnetized in the Voigt geometry, such as photonic crystals or HMMs studied here [Fig. 1(a)], nonreciprocity can be achieved for p-polarized light either through inhomogeneous magnetization¹⁵ or multilayered configuration.¹⁶ Here, we consider a multilayered HMM, which is easy to implement experimentally. The metamaterial is formed by periodically stacking unit cells consisting of a plasmonic (𝜖₂ < 0) layer sandwiched in between two dielectric (𝜖₁ $≠$ 𝜖₃, 𝜖₁ > 0, 𝜖₃ > 0) layers of thicknesses a_1,2,3, with the net lattice constant a₀ = a₁ + a₂ + a₃. The structure is subject to a DC magnetic field B₀ along the y-direction, and the plasmonic material is characterized by a dielectric permittivity tensor $𝜖 ^ 2 = [ 𝜖 2 , 0 , i Δ 2 ; 0 , 𝜖 2 , y y , 0 ; − i Δ 2 , 0 , 𝜖 2 ]$⁠,²⁰ where $𝜖 2 = 𝜖 ∞ − ω p 2 ω + i γ 2 − ω B 2 × 1 + i γ ω$⁠, $Δ 2 =i ω B ω × ω p 2 ω + i γ 2 − ω B 2$⁠, and $𝜖 2 , y y = 𝜖 ∞ − ω p 2 ω ( ω + i γ )$⁠, where 𝜖_∞ is the high-frequency permittivity, ω_p is the bulk plasma frequency, γ is the damping frequency, and $ω B = e m * B$ is the cyclotron frequency, where e and $m *$ are the charge and the effective mass of the electron, respectively. It is worth noting that strong nonreciprocity (large values of Δ₂) in metals requires strong DC magnetic fields on the order of several tesla.²⁰ Importantly, the nonreciprocal regimes reported here will also occur in HMMs made of highly doped semiconductors³⁶ in terahertz and infrared spectral domains for lower values of applied DC magnetic fields. The corresponding design of nonreciprocal hyperbolic metamaterials made of a doped semiconductor and operating in the mid-IR domain is presented in the supplementary material (Sec. II). Moreover, in structures made of ferrites,^12–19 the same effects can be achieved for magnetic fields of a few hundred Gauss, which is sufficient to provide saturation magnetization.

Starting with the exact transfer matrix technique, we develop an analytic effective medium theory with nonreciprocal corrections induced by the magnetization. For reciprocal structures (Δ₂ = 0), this procedure results in the well-known expression $K z 2 / 𝜖 ∥ + k x 2 / 𝜖 ⊥ = k 0 2$⁠, where $𝜖 ∥ = 𝜖 1 f 1 + 𝜖 2 f 2 + 𝜖 3 f 3$ and $𝜖 ⊥ = f 1 / 𝜖 1 + f 2 / 𝜖 2 + f 3 / 𝜖 3 − 1$ are the effective permittivities parallel and perpendicular to the layers, respectively, and f_m = a_m/a₀ is the volume fraction of the mth layer. By applying a similar procedure to the magnetized structure, we end up with two additional terms that are odd with respect to the wavenumber,

In the reciprocal case, such odd-order terms in k_x do not appear and the next term would be of fourth order, which is the direct consequence of the reciprocity k₀(k) = k₀(−k) [or ω(k) = ω(−k)]. This condition is clearly not satisfied for Eq. (1) where the two additional terms, linear and cubic with respect to k_x, lead to the nonreciprocal dispersion k₀(k) $≠$k₀(−k). One can see from Eq. (1) that these nonreciprocal terms increase with the MO parameter Δ₂ and/or the dielectric contrast between two layers adjacent to the magneto-plasmonic layer. The latter dependence confirms that for the inversion symmetric and bilayer structures (𝜖₃ = 𝜖₁), the nonreciprocity is not present.

The first- and third-order terms of the nonreciprocal contributions in Eq. (1) suggest that each one of them will dominate in different ranges of magnitudes of k_x. For small values of k_x $≪$ K_z, i.e., at the near normal incidence, the linear term in k_x on the right hand side will dominate. Its contribution is rather trivial and can be understood as a horizontal shift of the dispersion curves (either in elliptical or hyperbolic regimes) since $k x 2 −α k x ≈ k x − α / 2 2$⁠, where $α= Δ 2 𝜖 2 a 1 a 2 a 3 a 0 𝜖 ⊥ 𝜖 ∥ 𝜖 3 − 𝜖 1$ is a small parameter. For large values of k_x $≫$ K_z, the case of large wavenumbers that is of primary interest in the hyperbolic regime, the cubic term dominates instead.

It is instructive to mention that the nonreciprocal terms in Eq. (1) can also be described as magnetization-induced nonlocality⁴⁶ with the following k_x-dependent effective parameters:

The difference between this magnetization induced nonreciprocal nonlocality and the reciprocal nonlocality in conventional HMMs is that the effective parameters are odd functions of the wave-vector components.⁴⁶ 

Figure 1 shows the effect of the nonreciprocal corrections to the effective medium parameters $𝜖 ̃ ∥ k x$ and $𝜖 ̃ ⊥ k x$ for two opposite propagation directions, k_x = 15 × k₀ and k_x = −15 × k₀. As can be seen from Fig. 1(b), the effect of the nonreciprocal corrections, while being negligible for $𝜖 ̃ ∥ k x$⁠, is significant for $𝜖 ̃ ⊥ k x$⁠. Figure 1(b) shows that $𝜖 ̃ ⊥ k x$ has a pole near the wavelength λ = 360 nm, and the shape of the curve near this pole strongly depends on the sign of k_x, i.e., on the propagation direction. Another peculiarity occurs near λ = 450 nm, where $𝜖 ̃ ⊥$ exhibits additional variations which are also direction dependent. These are related to the epsilon-near-zero (⁠$𝜖 ∥ ≈0$⁠) condition and originate in the presence of $𝜖 ∥$ in the nonreciprocal terms in Eqs. (1) and (2). One of the most interesting consequences of nonreciprocity is that near such a pole the metamaterial can exhibit effective permittivities of the opposite sign for the two opposite propagation directions, i.e., $𝜖 ̃ ⊥ >$ 0 for the forward propagation k_x > 0, and $𝜖 ̃ ⊥ <0$ for the backward propagation k_x < 0, or vice versa, as illustrated in Fig. 1(c). The two hyperbolic regimes that occur in the structure, Type-II regime at longer wavelengths and Type-I regime at shorter wavelengths, are spectrally shifted for opposite directions of propagation. As a result, the onsets of the hyperbolic regimes for forward and backward propagating waves take place at different wavelengths.

To illustrate such nonreciprocal hyperbolic regimes, we first examine the metamaterial at the wavelength λ = 455 nm, which exhibits an elliptical regime ${ 𝜖 ̃ ∥ > 0 , 𝜖 ̃ ⊥ > 0 }$ in the absence of magnetization. Figure 2 shows changes in the dispersion as the off-diagonal component of the metal’s permittivity Δ₂ gradually increases, which is equivalent to an increase in magnetization. It can be seen from Fig. 2(a) that the isofrequency contours acquire progressively more asymmetric shapes for larger values of Δ₂, ultimately leading to a complete change in their topology.

This magnetization induced topological transition is more clearly revealed in Fig. 2(b), which shows how one side of the closed elliptical contour (k_x < 0) gradually changes to an open Type-II hyperbolic contour ${ 𝜖 ̃ ∥ < 0 , 𝜖 ̃ ⊥ > 0 }$⁠, while the opposite side (k_x > 0) of the contour remains elliptical.

Next, we will study the effect of magnetization on the metamaterial at the wavelength λ = 360 nm, which in the absence of magnetization corresponds to a Type-I hyperbolic regime ${ 𝜖 ̃ ∥ > 0 , 𝜖 ̃ ⊥ < 0 }$⁠. Figure 2(c) shows that as in the previous case, the contours become progressively more asymmetric in shape as Δ₂ increases. In particular, the left-side of the hyperbola (k_x < 0) opens wider, but maintains its original topology. The right side (k_x > 0), in contrast, experiences a topological transition from an open hyperbola to a closed ellipse, which eventually collapses to a point when the MO parameter reaches a critical value of Δ_cr $≡$ Δ₂ ≈ 0.04𝜖₂, beyond which only the backward propagation is possible. This magnetization induced topological transition is further illustrated in Fig. 2(d): the right hyperbolic contour k_x > 0 gradually morphs into the isolated ellipse (tube), which then collapses to a single point and finally disappears leading to the one-way hyperbolic regime.

To summarize, we demonstrate and identify three distinct nonreciprocal hyperbolic regimes. The first regime—the nonreciprocal two-way hyperbolic regime—is characterized by the contours consisting of two asymmetric hyperbolas (for both k_x < 0 and k_x > 0), topologically equivalent to that of the nonmagnetic Type-I structure [Fig. 2(c)]. The second regime—the forward-elliptical and backward-one-way hyperbolic regime—is characterized by a hyperbola for k_x < 0 and an ellipse for k_x > 0. This regime can be subdivided into two classes corresponding to Type-I or Type-II hyperbolic regime. In the case of Type-I hyperbolic regime, in addition to the hyperbolic branch, we encounter a closed and isolated ellipse of an asymmetric shape [Fig. 2(c)], whereas for Type-II, there is half of an ellipse connected to a hyperbola [Fig. 2(a)]. The third regime—the complete one-way hyperbolic regime—appears with the further increase of the magnetization for the Type-I HMM [Fig. 2(c)] when the ellipse corresponding to k_x > 0 collapses.

Its important to mention here is that while in our calculations we used particular thickness of the layers to be 14/20/14 nm in the unit cell, due to the fact that all the dimensions are much smaller than the wavelength of light and electromagnetic waves perceive the hyperbolic metamaterial as homogenized, the response is expected to be tolerant to deviations from the precise thicknesses in both transverse and longitudinal directions. Indeed, the growth of multiple layers with nanoscale precision is not easily feasible, and the layers typically vary in their thickness both within the same layer and among different layers within the stack. However, as long as we can ensure the desirable volume fraction on average, the response of the metamaterial will remain the same, which has also been evidenced by the experimental studies.^43,44

The hyperbolic dispersion originates from the coupling of surface plasmon polaritons (SPPs) supported by individual plasmonic layers comprising HMMs. Nonreciprocal hyperbolic regimes described here have the same origin, with the difference that the coupling takes place between surface magneto-plasmons, i.e., surface plasmons whose dispersion is modified by magnetization. To further reveal the origin of nonreciprocal and one-way hyperbolic dispersion predicted by the effective medium theory equations (1) and (2), and to make predictions for realistic structures, we will now examine the eigenmodes and the transmission spectra calculated with the exact transfer matrix technique for a finite layered system.

Natural plasmonic modes of layered structures are known to manifest themselves as poles in the transmission spectra and can clearly be seen in Figs. 3(a) and 3(b). Figure 3(a) shows the case of a single metal film (one unit cell of the HMM), where such poles form two continuous (low-frequency and high-frequency) dispersion curves corresponding to two magneto-plasmons that are predominantly localized on opposite metal-dielectric interfaces. These modes are separated by a frequency gap originating from the asymmetric cladding of the metal layer (𝜖₁ $≠$ 𝜖₃). Note that a similar gap also exists in the dispersion of the so-called “short-range” and “long-range” plasmons in structures with an inversion symmetric unit cell. However, in the latter case, the gap originates from the coupling of the plasmons on two opposite interfaces. In the case of the asymmetric cladding considered here, such coupling between plasmons still exists, but is significantly suppressed due to the mismatch in their eigenfrequencies. More importantly, the dispersion of the magneto-plasmons exhibits nonreciprocity λ(k) $≠$ λ(−k), and in contrast to the non-magnetic case, the curves approach different asymptotes (indicated by dashed horizontal lines) for the forward (k_x > 0) and backward (k_x < 0) propagating waves. As it is shown below, this nonreciprocity of the magneto-plasmons of the individual unit cell is the source of the nonreciprocal hyperbolic regimes found in multilayered structures.

As the next step, we consider a structure consisting of 10 unit cells with its magneto-plasmonic bands plotted in Fig. 3(b). As expected, the number of modes increases and the modes tend to push each other to the domains of longer wavenumbers and shorter wavelengths, indicating onset of the hyperbolic regimes. One can immediately establish a correspondence between the results of the effective medium theory outlined above and the exact calculations presented in Fig. 3(b). Thus, the Type-I (Type-II) hyperbolic regime occurs due to hybridization of the high frequency (low frequency) surface magneto-plasmons of the individual metal layers. The short-wavelength Type-I and long-wavelength Type-II regimes appear to be separated by the region of elliptical dispersion. Added to this, in agreement with the effective medium calculations, the modes appear to be strongly nonreciprocal. It is worth noting that despite the hybridization of the magneto-plasmons of individual metal layers, all of the modes of multilayered structures are still approaching the same asymptote as in the case of the single unit cell. As a result, the corresponding hyperbolic bands have different “cut-off” wavelengths for the opposite propagation directions, explaining the origin of one-way hyperbolic regimes. In particular, there is a frequency window, from λ ≈ 450 nm to λ ≈ 480 nm, where Type-II one-way hyperbolic regime is realized, and another window from λ ≈ 350 nm to λ ≈ 365 nm where Type-I one-way hyperbolic regime occurs.

From subwavelength resolution to enhanced lasing efficiency, the hyperbolic dispersion enables many fascinating applications, all made possible by the presence of plasmonic modes with very long wavenumbers.^34–44 Another advantage of HMMs is the non-resonant origin of their unique response, which amounts to the broadband character of the hyperbolic regime. This makes the HMMs very promising for various applications where broadband characteristics are required. Similar arguments can apparently be applied to nonreciprocal photonic devices. Indeed, the broadband nonreciprocal response can be achieved only in bulky optical components. To our best knowledge, all attempts to reduce the footprint of nonreciprocal devices to make them more compatible with the contemporary integrated photonic components have so far relied on the use of resonant effects.^18–26 While resonances do allow enhancement of the nonreciprocal response, they also significantly reduce the operational bandwidth of the devices. Here, we show that the nonreciprocal HMMs do not have this limitation and may offer nonreciprocal response over a broad operational bandwidth.

Figure 4(a) shows transmission through the metamaterial (10 unit cells) in the Type-II hyperbolic regime for forward and backward propagation. A nonreciprocal transmission indeed occurs in the broad spectral window defined by the offset Δω of the forward and backward transmission bands. The bandwidth of one-way response is defined by the difference in the cut-off frequencies for the forward and backward hyperbolic transmission bands in Fig. 3, which, in turn, is defined by the strength of magnetization. Thus, the bandwidth of the one-way response is only limited by the strength of the magnetic field and MO response of the materials constituting the structure, which is in sharp contrast to the bandwidth-limited nonreciprocal devices based on resonant MO structures. In resonant structures, the optical isolation (one-way response) relies on MO-induced splitting of a narrow resonance of bandwidth Γ for forward and backward propagation directions, by the extent that the splitting Δω exceeds the bandwidth (Δω > Γ).¹⁶ As a result, Lorentzian-shaped one-way transmission occurs over Γ-wide band. In contrast to this principle of operation, the nonreciprocal HMM provides a nearly uniform transmission over the entire frequency range Δω [Fig. 4(a)]. As with a sufficient number of layers the hyperbolic transmission bands can be made arbitrarily wide, one can always design a nonreciprocal device with the maximally possible band- width Δω.

The origin of the nonreciprocity can also be understood from the electric field distribution inside the HMM. The field distribution, calculated for the wavelength λ = 470 nm, is plotted in Fig. 4(b) for forward (k_x = 7k₀) and backward (k_x = −7k₀) propagation directions. The transmission in the forward direction occurs due to the excitation of the side-coupled surface plasmons (on the left side of metallic layers) propagating along the layers in the forward direction (k_x > 0), which transfer the electromagnetic energy through the structure. On the other hand, when excited from the opposite side with the backward propagation direction (k_x < 0), the excitation wavelength exceeds the cut-off wavelength (λ₀ = 480 nm) and no plasmonic modes are excited, resulting in the fast decay of the field inside the structure and vanishingly small transmission.

As for any other plasmonic structure, Ohmic losses will play a detrimental role for operation of the nonreciprocal HMM. For example, Fig. 4(a) shows how the transmission through the metamaterial changes with the increase of the damping frequency from that in the epitaxially grown silver to the thermally evaporated one.⁴⁸ As can be seen from Fig. 4(a), while the bandwidth of the nonreciprocal response stays nearly unchanged or even decreases, the transmission drops. To avoid this decrease in the transmission, the number of layers in the HMM should be reduced as shown in the supplementary material (Sec. III). However, this may also lead to the narrowing in the bandwidth of the hyperbolic transmission band. Therefore, in addition to the strength of the MO response, the limitation in the operational bandwidth of nonreciprocal HMM devices will also be dictated by losses. Nevertheless, it is apparent that the operational bandwidth of the HMM can always be made superior to that of resonant plasmonic structures where losses have even more detrimental effects (see Sec. III of the supplementary material). Moreover, the field profiles, Fig. 4(b), suggest another class of applications which are possible even for strongly absorbing structures, such as nonreciprocal and one-way absorbers. The magneto-plasmons in the nonreciprocal HMMs are excited only for one direction, which implies that the absorption of incident wave will take place only in this particular direction, while a complete reflection will occur for the other.

We demonstrated the possibility of nonreciprocal light transmission using magnetoplasmonic hyperbolic metamaterials. New nonreciprocal hyperbolic regimes and one-way topological transitions between hyperbolic and elliptical dispersion regimes were revealed. Thanks to the non-resonant nature of the metamaterial, previously unachievable broadband nonreciprocal transmission was demonstrated which makes this design principle promising for practical applications. In addition to the visible domain studied here, the results presented hold a great potential for applications at the infrared and terahertz frequencies where nonreciprocal hyperbolic metamaterials are made of highly doped semiconductors.

See supplementary material, which includes Ref. 46, for details of analytic theory.

This work was supported by the National Science Foundation (Nos. CMMI-1537294 and EFRI-1641069).