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Abstract
We report dynamic regulation of nonlocal degree, nonlocal effects and spatial dispersion characteristics for transverse electric (TE) waves in periodic layered gyromagnetic metamaterials (PLGMs) by an applied magnetic field. A nonlocal effective permeability tensor, relying on both frequency and wave vector, is derived by expanding the accurate dispersion relation obtained by the transfer-matrix method (TMM) to high-order terms. The numerical results indicate that the degree of nonlocality of electromagnetic response in such PLGMs is closely dependent on the ratio between the period of PLGMs and the working wavelength. There are giant spatial nonlocality and strong spatial dispersion near the center or boundary of the first Brillouin zone, which leads two or three propagating modes to appear in these regions for a fixed frequency. Interestingly, the degree of nonlocality, nonlocal effects and spatial dispersion properties in such PLGMs can be manipulated dynamically by an applied static magnetic field. In addition, it is possible that a quasi-straight isofrequency contour occurs in the case of linear response. These properties make the PLGMs become excellent candidates for designing photonic devices in information communication, storage, nondiffraction transmission, and so on.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
With the progress of modern technology, natural materials are increasingly difficult to meet the development of information society. Therefore, researchers turn their attention to artificially synthetic materials with specific functions [1]. Photonic crystals [2–4] are a kind of periodic artificial microstructures proposed earlier, whose period is comparable to the wavelength of incident light, and have exhibited many novel optical properties, such as photonic bandgap and localization of photons [5,6].
At present, researchers are turning their attention to electromagnetic metamaterials [7]. Electromagnetic metamaterials can be regarded as a kind of homogenous effective medium with the local effective permittivity and permeability components described by the local effective medium theory (LEMT), when their periods are much less than the incident wavelength [8] . This kind of homogenous metamaterials have exhibited many exotic electromagnetic properties, such as negative refraction [9–12] and sub-wavelength imaging [13], hyperlensing [14], optical cloaking [15–17], self-induced torque [18,19], spatial filtering [20], nondiffraction transmission [21], and so on.
However, the electromagnetic properties of metamaterials in the range of $0.1\;<\;d/\lambda\;<\;1$, where $d$ is the period scale and $\lambda$ is the wavelength of incident electromagnetic waves, have not been fully revealed. On the one hand, the LEMT may no longer be applicable in this region, and the nonlocal effective medium theory (NEMT) should be adopted [8]. This is because that the field distribution can not be considered uniform in this range, and the coupling between adjacent cells is stronger, which leads the nonlocality of electromagnetic response to be considered necessarily [22] . Otherwise, the electromagnetic properties of materials can not be objectively reflected. On the other hand, in this region, spatial nonlocal effects and spatial dispersion characteristics will be revealed, new propagating modes and new physical phenomena will appear [23].
The so-called nonlocal electromagnetic response is that the polarization or magnetization at a spatial point is induced by fields not only at the same point, but also at other positions in the vicinity of the spatial point. Consequently, dielectric permittivity or magnetic permeability of nonlocal materials is not only the function of frequency, but also the function of wave vector in spectral domain [24,25]. Since Pekar pioneered spatial nonlocality in semiconductor materials [26], the nonlocal effects in semiconductor nanostructures, plasmon and metamaterials have attracted much attention, giving rise to the development of nonlocal theory [27–29].
Metamaterials have shown strong spatial nonlocality and spatial dispersion properties due to the surface plasmon existing at the interface between two mediums with oppositely signed permittivity $\epsilon$ or permeability $\mu$ [30,31]. There are two kinds of surface plasmons, namely electric surface plasmon (ESP) [32,33] and magnetic surface plasmon (MSP) [34] . ESP is a charge-density wave, propagating along the interface of metal-dielectric structure [35]. However, MSP originates from the effective magnetic surface charges participating in coupled electromagnetic modes, which can appear in the gyromagnetic metamaterials [36,37].
In this paper, we investigate primarily spatial nonlocal effects and spatial dispersion properties of transverse electric (TE) waves in periodic layered gyromagnetic metamaterials (PLGMs) with an external magnetic field. Our purpose is to clarify mainly how the nonlocality of electromagnetic response of the PLGMs depends on the ratio, $d/\lambda$, and how to control the nonlocality and spatial dispersion properties of the PLGMs dynamically by an applied magnetic field, and find some novel physical phenomena simultaneously.
This paper is organized as follows: In Sec. 2, the nonlocal effective medium model for the PLGMs is proposed by expanding the accurate dispersion relation to fourth order terms, from which the nonlocal effective permeability tensor is derived. In Sec. 3, the spatial nonlocality and spatial dispersion properties depending on the ratio $d/\lambda$ are analyzed in detail. In addition, dynamic regulation of nonlocal effects and nonlocal degree in the PLGMs by an applied magnetic field is demonstrated, and a kind of quasi-straight isofrequency contours (IFCs) are found and their potential applications are elaborated. Finally, a conclusion of the present study is given in Sec. 4.
2. Nonlocal effective medium model
2.1 Accurate dispersion relation
As illustrated in Fig. 1, we consider a periodic multilayered metamaterial consisiting of alternating arrangement of dielectric and gyromagnetic layers, which is called periodic layered gyromagnetic metamaterials (PLGMs). The gyromagnetic layer is the low-loss gyromagnetic material which consists of the yttrium-iron-garnet (YIG) [38]. The thickness of the dielectric and gyromagnetic layers in a unit cell is $d_{1}$ and $d_{2}$, respectively. The permittivity and permeability of the isotropic dielectric layer are denoted as $\varepsilon _{1}$ and $\mu _{1}$, respectively. While they are denoted as $\varepsilon _{2}$ and $\widehat {\mu }_{2}$ in the gyromagnetic layer. In particular, $\widehat {\mu }_{2}$ is a permeability tensor with the following form [39]
Fig. 1. (a) Schematic of a TE wave incident on the PLGMs consisting of alternating layers of an ordinary dielectric and a gyromagnetic material, with the thickness $d_{1}=1$ mm, $d_{2}=2$ mm and the permittivity $ \varepsilon _{1}=2.25,\;\varepsilon _{2}=15$, respectively. $k_{i}$, $k_{r}$ and $k_{t}$ are wave vectors of incident, reflective and refractive waves, respectively. (b) Geometry of the PLGMs.
Fig. 2. The effective permeability $ \mu _{e}$ of the gyromagnetic layer versus the ratio $d/ \lambda$ for different applied magnetic fields.
In the absence of the applied static magnetic field, namely $H_{0}=0$, we have $\mu _{a}=1$, $\mu _{b}=0$, and $\mu _{e}=1$. Then the dispersion relation (Eq. (2)) of the PLGMs degenerates to
2.2 Nonlocal effective permeability tensor
We assume that a monochromatic plane wave incidents on the PLGMs with the TE polarization along $z$ axis, having the form $\mathbf {E}=\hat {\mathbf {z}} E_{0}e^{i(k_{x}x+k_{y}y-\omega t)}$, as shown in Fig. 1. The electromagnetic properties of the PLGMs can be described by an effective medium theory in the long wavelength limit, if the period of the PLGMs is much smaller than the wavelength, that is $d\ll \lambda$. In this situation, the permittivity and permeability of the PLGMs are characterized by two tensors of the forms:
3. Dynamic control of nonlocal effect and spatial dispersion
3.1 The ratio $d/ \lambda$
In the following, we will show that the nonlocality of the PLGMs response to TE waves is dependent on the ratio $d/\lambda$. Particularly, the degree of nonlocality, nonlocal effects and spatial dispersion properties can be manipulated dynamically by the applied static magnetic field $H_{0}$. In order to elaborate these properties, we apply TMM, LEMT and NEMT to analyze the nonlocality of the PLGMs by contrast, where the LEMT is based on the Ref. [40]. Figure 3 shows the dispersion curves of TE modes in the PLGMs without an external magnetic field, namely $H_{0}=0$ [Fig. 3(a)], and with an external magnetic field, $H_{0}=0.4$ T [Fig. 3(b)] in the case of $k_{x}=0$. Figures 3(c) and 3(d) are two enlarged diagrams of Fig. 3(b) for the regions ($0.128\;<\;d/\lambda\;<\;0.16$). In the absence of the external magnetic field, the PLGMs become a common metamaterial formed by alternating arrangement of two isotropic nonmagnetic mediums. In this case, there exists only one branch of dispersion curve with a positive slope, namely one forward wave when $0\;<\;d/\lambda\;<\;0.27$ [see red lines in Fig. 3(a)], obtained by the exact dispersion relation (Eq. 2) on the basis of the TMM. When $d/\lambda\;>\;0.27$, however, there exist two or multiple branches with a positive slope, which indicates it is possible that two or multiple forward waves occur in the metamaterials for certain frequencies, as shown in Fig. 3(a). In other words, the spatial nonlocality of electromagnetic response and nonlocal effects begin to appear when $d/\lambda\;>\;0.27$, which leads additional modes to occur. These features are similar as in a nonmagnetic metamaterials of only dielectric response [30].
Fig. 3. Dispersion diagrams of TE waves in the PLGMs as a function of $k_{y}d/ \pi$ at $H_{0}=0$ (a), and $H_{0}=0.4$ T (b) with $k_{x}=0$, based on the three kinds of different theories: TMM (red lines), LEMT (blue lines), NEMT (black lines), respectively. (c) and (d) are two enlarged diagrams of (b) in the range of $0.128\;<\;d/ \lambda\;<\;0.16$.
In the present paper, we use an external magnetic field to control the dispersion characteristics of gyromagnetic metamaterials. It is instructive to see that the external magnetic field has obvious influence on the dispersion characteristics. When $H_{0}=0.4$ T, the dispersion characteristics are significantly different from the case of $H_{0}=0$ in the range of $0\;<\;d/\lambda\;<\;0.2$, as shown in Fig. 3(b). From Fig. 3(b), it can be seen that the rightmost dispersion curve breaks and forms a band gap [see Fig. 3(c)] within a certain ratio range ($0.137\;<\;d/\lambda\;<\;0.152$). When $H_{0}=0$, however, there isn’t any band gap. It is also remarkable that an abnormal dispersion branch appears in the vicinity of $d/\lambda =0.13$ [see Fig. 3(d)], which has a dip. Apparently, the group velocity of the wave is zero at the dip when $d/\lambda =0.1298$. In addition, there are three different wave-numbers $(k_{y})$ for a fixed frequency [see points $1,2,3$ on the red lines in Fig. 3(d)]. Interestingly, point $1$ corresponds to an additional wave with the negative group velocity, which isn’t existent in the case of $H_{0}=0$, as shown in Fig. 3(d). The points $2$ and $3$ correspond to forward waves with the positive group velocity. In other words, a backward wave and two forward waves coexist at the same frequency. Compared to the TMM, the LEMT can only predicts one branch and exhibit obvious deviation from accurate dispersion curves when $d/\lambda$ gets greater. Therefore, we should use the theory of nonlocal effective medium to analyze the electromagnetic properties of metamaterials when $d/\lambda$ does not satisfy this condition that $d/\lambda\;<<\;1$.
3.2 The external magnetic field $H_{0}$
In order to elaborate the regulatory law of the applied static magnetic field on the degree of nonlocality, nonlocal effects and spatial dispersion properties for TE modes in the PLGMs. Figure 4 shows the dispersion properties of TE modes as a function of $H_{0}$ and $k_{y}$ with $k_{x}=0,$ $d/\lambda =0.131$ and $f_{1}=1/3$. From Figs. 4(a) and 4(d), it can be seen that for the small or large magnetic field ($0\leq H_{0}\leq 0.325$ T, or $H_{0}\geq 0.5$ T), there is only one branch of dispersion curve (namely only one mode), and the dispersion curves, obtained by the TMM, NEMT and LEMT, are also basically coincident. In this situation, the PLGMs can be considered to be a local medium when the magnetic field is in the two ranges. However, when $0.325$ $\mathrm {T}\;<\;H_{0}\;<\;0.38$ T, as shown in Fig. 4(b), a band gap occurs, whose width increases as the angle of incidence increases. These characteristics can be widely used in the design of switches and wave cut-off devices. Furthermore, when $0.39$ $\mathrm {T}\;<\;H_{0}\;<\;0.404$ T, there are multiple branches of dispersion curve (namely multiple modes), which indicates that the nonlocality of electromagnetic response of the PLGMs is enhanced, and nonlocal effects are more evident and stronger. The nonlocality of the PLGMs arises from the magnetic surface plasmon (MSP) at the interfaces between the gyromagnetic and dielectric layers, caused by the variation of the electromagnetic field on the periodic scale. Therefore, the nonlocality result in the appearance of three propagating modes. From the above discussion, we can see that for a special ratio $d/\lambda$, the nonlocality of the PLGMs can be dynamically regulated by the external magnetic field.
Fig. 4. The manipulation of $H_{0}$ on the dispersion diagrams of the TE wave in the PLGMs of the ratio $d/ \lambda =0.131$ and $k_{x}=0$.
It is well known that the propagating properties of electromagnetic waves can also be explicitly analyzed by isofrequency contours (IFCs). Figure 5 displays the IFCs of TE modes based on Eq. (2) for three different ratios: (a) $d/\lambda =0.09$, (b) $d/\lambda =0.1301$, (c) $d/\lambda =0.131$, with $H_{0}=0.4$ T. From Fig. 5, we can see that the variation of $d/\lambda$ can cause a significant change of IFCs. When $d/\lambda =0.09$, the IFCs is only a circular, as shown in Fig. 5(a). If $d/\lambda =0.1301$, however, analogous parabolic and quasi-straight IFCs appear, as shown in Fig. 5(b). It should be noted that the IFCs consist of an ellipse, a pair of analogous parabola and a pair of quasi-straight curves when $d/\lambda =0.131$ in Fig. 5(c), where the elliptic isofrequency contour corresponds an additional mode, whose emergence is a prominent feature of giant nonlocality and strong spatial dispersion. Obviously, the nonlocal effect and spatial dispersion are strong in the vicinity of the center of the first Brillouin zone. Therefore, we can control the spatial dispersion of the PLGMs by the ratio $d/\lambda$.
Fig. 5. The IFCs based on Eq. (2) at $H_{0}=0.4$ T with three different ratios: (a) $d/ \lambda =0.09$; (b) $d/ \lambda =0.1301$; (c) $d/ \lambda =0.131$, respectively.
In Fig. 6, we plot the IFCs based on Eq. (2) for three different external magnetic fields: (a) $H_{0}=0.43$ T, (b) $H_{0}=0.44$ T, (c) $H_{0}=0.45$ T, with $d/\lambda =0.143$. We can observe from Fig. 6, as $H_{0}$ increases, the IFCs have changed dramatically for the given structure and incident frequency. Figure 6(a) shows the dispersion curve of parabola-like at $H_{0}=0.43$ T, which illustrates only a refracted wave exists. Figure 6(b) reveals general elliptic and parabola-like dispersions at $H_{0}=0.44$ T simultaneously. It is worth noting that the IFCs are a combination of an ellipse, a pair of parabola-like and a pair of flat curves at $H_{0}=0.45$ T in Fig. 6(c), where the elliptic IFCs correspond to an additional mode. Therefore, the spatial dispersion properties of the PLGMs can be manipulated by the external magnetic field.
Fig. 6. The IFCs based on Eq. (2) for the PLGMs of the ratio $d/ \lambda =0.143$, with three different external magnetic fields: (a) $H_{0}=0.43$ T; (b) $H_{0}=0.44$ T; (c) $H_{0}=0.45$ T, respectively.
3.3 Dispersion characteristics
In addition, the IFCs, in the vicinity of the boundary of the first Brillouin zone, also exhibit strong spatial dispersion characteristics for a particular structure and incident frequency. In order to explain this point of view, the IFCs calculated by the LEMT with Eq. (6) (red curve), the NEMT with Eq. (10) (green curves) and the TMM with Eq. (2) (black curves), are displayed in Fig. 7(a). By the accurate IFCs (black curves), we can see that there are two possible propagating modes near the boundary of the first Brillouin zone, and the NEMT can also well predict this situation, namely two modes. However, the LEMT can only predict one mode. In addition, comparison with the NEMT and the TMM, the IFCs obtained by the LEMT are significantly different from them.
Fig. 7. (a) IFCs based on LEMT (red line), NEMT (green lines) and TMM (black lines) with $d/ \lambda =0.131,H_{0}=0.5$ T. (b) The three-dimensional band structures of the PLGMs at $H_{0}=0.4$ T with $k_{x}$ spanning the first Brillouin zone.
Figure 7(b) shows the band structure of the PLGMs in the three-dimensional $\omega -k$ space, according to Eq. (2) with the nonlocality. It can be seen that there exists a Dirac cone at the band structure, and the Dirac point is located at $\mathbf {k}=0.$ In the vicinity of the Dirac point, giant nonlocality and strong spatial dispersion properties occur, which leads additional modes to appear. In other words, an incident beam may split into multiple beams when it is incident on the PLGMs, near the Dirac point due to the nonlocal effects in the PLGMs.
3.4 Nonlocal effective permeability components
Next, we study how the nonlocal effective permeability components ($\mu _{x}^{N,\;eff}$ and $\mu _{y}^{N,\;eff}$) rely on the wave vector components $k_{x}d/\pi$, $k_{y}d/\pi$ and the magnetic field $H_{0}$ in the specific frequency $\omega =91.73$ $\mathrm {GHz}$, based on the NEMT, as shown in Fig. 8. From Fig. 8(a), we can see that when $0\;<\;H_{0}\;<\;0.38$ T, $\mu _{y}^{N,\;eff}$ is always positive and it increases to positive infinity as $k_{x}d/\pi$ tends to $1$. However, $\mu _{x}^{N,\;eff}$ changes from negative to positive infinity at $k_{y}d/\pi\;<\;0.5$, but $\mu _{x}^{N,\;eff}\;>\;0$ at $k_{y}d/\pi\;>\;1$ in Fig. 8(b). In this process, the dispersion curves of the PLGMs are parabolic-like or elliptical-like. Hence, the PLGMs shows a weak nonlocal effect in this range. For $0.38$ $\mathrm {T}\;<\;H_{0}\;<\;0.65$ $\mathrm {T}$, $\mu _{y}^{N,\;eff}\;<\;0$, but $\mu _{x}^{N,\;eff}$ exhibits great changes, which decreases from zero to negative infinity, then becomes positive infinity and reduces to zero with $H_{0}$ increasing for small $k_{y}$. Meanwhile $\mu _{x}^{N,\;eff}$ is always negative for large $k_{y}$. Obviously, this is the key feature of spatial dispersion [30]. When $H_{0}\;>\;0.65$ T, $\mu _{y}^{N,\;eff}\;>\;0$ for any $k_{x}$, and $\mu _{x}^{N,\;eff}\;<\;0$ for small $k_{y}$, leading to parabolic-like dispersion curves, and $\mu _{x}^{N,\;eff}\;>\;0$ for large $k_{y}$, inducing elliptical-like dispersion curves.
Fig. 8. Nonlocal effective permeability tensor components (a) $ \mu _{y}^{N,\;eff}$, (b) $ \mu _{x}^{N,\;eff}$ as functions of normalized wavevector components, $k_{x}d/ \pi$, $k_{y}d/ \pi$, and the external magnetic field $H_{0}$ when the ratio $d/ \lambda =0.146$.
3.5 Refraction of TE waves in PLGMs
We can use IFC to analyze the beam refraction behavior at the interface between the free space and the PLGMs. By the IFCs, one can understand the character of refracted waves and the general relationship between the directions of energy flow (group velocity) and phase velocity of refracted waves within the PLGMs [41]. Based on the definition of group velocity, $\mathbf {v}_{g}={\triangledown} _{\mathbf {k}}\omega (\mathbf {k}),$ the group velocity must be oriented normal to the IFCs. There are two possible normals to the isofrequency contour, pointing "inwards" and "ourwards". By calculating $\mathbf {v}_{g}$, we can determine which of the two possible normal directions yields increasing frequency, and then the correct direction of group velocity is determined [42]. In addition, the energy flow of the transmitted waves must be away from the interface. Therefore, the normal component of the group velocity, $\mathbf {v}_{gy}$, must have the same sign on both sides of the interface. Thus, we choose the transmitted wave to be the one with negative $x$ component of $\mathbf {v}_{g}$ in Fig. 9. Figure 9 shows the three typical cases of IFCs and the beam refraction diagrams, corresponding to the different frequency and the external magnetic field: (a) $d/\lambda =0.143,$ $H_{0}=0.43$ T; (b) $d/\lambda =0.13,$ $H_{0}=0.4$ T; (c) $d/\lambda =0.1298,$ $H_{0}=0.4$ T. In the first case [see Fig. 9(a)], there exists only one refracted beam, whose group velocity is negative refraction, while the wave vector is positive refraction. In particular, there are three refracted beams from the splitting of the incident beam due to the spatial nonlocal effects in the PLGMs. Among of them, one beam has the elliptical isofrequency contour (black curve) with the negative refraction of both the group velocity and the wave vector. The second wave possesses parabola-like dispersion (red curve), whose group velocity is negative refraction, but the wave vector is positive refraction [indicated in Fig. 9(b)]. It should be noticed that the third branch (blue curve) in Fig. 9(b) approaches to a straight line, consequently the beam with this kind of flat dispersion contour has the characteristics of approximate nondiffraction propagation in the case of linear response. Note that there exists still another interesting case, namely only one straight isofrequency contour occurs when the incident beam is near vertical incidence, as shown in Fig. 9(c). Therefore, the beam propagation in the PLGMs can be manipulated by the ratio $d/\lambda$ and the applied static magnetic field $H_{0}$.
Fig. 9. Schematic diagrams of group velocity and wavevector directions inside the PLGMs for the three kinds of cases: (a) $d/ \lambda =0.143,H_{0}=0.43$ T, (b) $d/ \lambda =0.13,H_{0}=0.4$ T, (c) $d/ \lambda =0.1298,H_{0}=0.4$ T. $k_{i},\;k_{t}$ are wave vectors of incident and refractive waves, respectively. $\mathbf {v}_{g}$ is group velocity vector. The incident wave comes from air and has a TE polarization.
4. Conclusion
We have investigated the nonlocal effects and spatial dispersion properties of TE waves in the PLGMs by employing the TMM, LEMT and NEMT comparatively, with the analyses of the dispersion diagrams and the IFCs. We have shown that the nonlocal degree of electromagnetic response of the PLGMs is closely dependent on the ratio between the period of PLGMs and the working wavelength. We have also demonstrated the degree of nonlocality, nonlocal effects and spatial dispersion properties in such PLGMs can be manipulated dynamically by an applied static magnetic field. In addition, by adjusting the applied magnetic field, this kind of gyromagnetic metamaterials can undergo interesting transitions from local response to nonlocal response, and vice versa. Some novel physical phenomena, such as the coexistence of three propagating modes due to the spatial nonlocal effects and the emergence of a flat dispersion curve for a fixed frequency, are demonstrated, opening up new information channels. These properties have extensively potential applications in information communication, storage, nondiffraction transmission and other fields.
Appendix
The nonlocal effective permeability components $\mu _{x}^{N,\;eff}$ and $\mu _{y}^{N,\;eff}$ in Eq. (6) for the NEMT can be calculated as follows. First of all, we expand Eq. (2) into power series of $k_{x}d$, $k_{1x}d1$ and $k_{2x}d2$ up to fourth-order terms, respectively. The expanded equation is expressed as
Funding
Natural Science Foundation of Guangdong Province (2016A030313439, 2018A030313480); Key Program for Guangdong NSF of China (2017B030311003); Science and Technology Program of Guangzhou City, China (201707010403).
Disclosures
The authors declare no conflicts of interest.
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