Introduction

Metamaterial is used to achieve the controls of electromagnetic properties which go beyond the limitations of natural materials [1–5], where a large number of fascinating physical phenomena can occur [6–9], such as negative-index materials (NIMs) and zero-index materials (ZIMs). In 1968, Veselago theoretically predicted that NIMs with $n=-\sqrt{\epsilon \mu }$, where the permittivity $\epsilon$ and the permeability $\mu$ are both negative, would exhibit unusual properties [10]. In a positive-index material (PIM), the triplet set of vectors $E, H, k$ where E is the electric, H the magnetic, and k the wave vector, is a right-handed set, whereas in a NIM it is a left-handed one. In such NIMs, the direction of the group velocity $v_g$ is antiparallel to the phase velocity $v_p$. Veselago also showed that light refracted at an interface between a PIM and a NIM stays on the same side of the normal as the incident light. Nearly three decades later, J. B. Pendry came up with a setup for negative index response [11,12] and D. R. Smith numerically and experimentally actualized negative refraction adopting periodic continuous metallic wires and periodic split ring resonators (SRRs) for the first time, which confirmed effective negative permittivity and permeability respectively [13,14]. Subsequently, numerous experimental and theoretical studies are focused on realizing NIMs using different artificial periodic structures [15–23], for example, SRRs [17], transmission lines [18], photonic crystals [19], permanent magnetic ferrites [20], closed ring magnetic dipole resonators [21], parallel nanorods [22], and S-shaped resonator [23]. Due to its potential in subwavelength imaging and focusing beyond the diffraction limit [24, 25], negative refraction has attracted much attentions. The same situation also occurred on ZIMs. Recently, researchers have shown increasing interest in ZIMs [26, 27], because of their unique properties and applications, including electromagnetic energy squeezing through narrow waveguide channels [28], radiation patterns manipulating [29], infinite phase velocity and zero phase accumulation [30, 31].

Although NIMs and ZIMs have great prospects for future applications, they also cause some controversies in physics, one of which is whether causality and the speed limit c are broken [32]. A simple diagram shows this situation in Fig. 1(a). When wave refracts at the interface between PIM and NIM, the wavefront AB transfers to AD, which means the transmission process B-C-D finishes immediately. Therefore, point B reaches point D at an infinite speed exceeding the speed limit c, in other words, causality is violated (as described in Fig. 1(b) of Ref [32].). The reason why the controversy appears is that previous researchers mainly focused on the researches of steady state but they ignored that some time is needed to complete the establishment of negative refraction. Reference [33] takes a good step in solving the controversy, where a specific photonic crystal NIM was used to study the transient establishment of negative refraction, thus explaining the occurrence of negative refraction without causality or speed limit violation. It is noted that transient analysis is necessary for handling the light propagation problems in materials, especially in the case of novel NIM and ZIM. However, so far, there are no previous study that has dealt with the transient process for zero refraction. Furthermore, no systematic investigation is presented comparing the establishment processes of negative, zero and positive refraction.

 

Fig. 1 (a) Schematic diagram demonstrates the supposed breakdown of causality or the violation of speed limit c as described in Ref [32]. (b) A wedge-shaped structure stacking by the unit cell shown in Fig. 1(c). The incident wave propagates along the x axis, shooting normally onto the first interface of the wedge. (c) Schematic diagram of a unit cell consisting of one SRR and one metallic wire deposited on opposite sides of a 0.25-mm-thick substrate ($\epsilon =4.4$, loss tangent of 0.02). The propagation of the electromagnetic field is along the x axis, the electric field is oriented along y axis, and the magnetic field is along the z axis. Specific parameters: a_x = 2.5 mm, a_y = 2.5 mm, a_z = 2.5 mm, W = 0.14 mm, L = 2.2 mm, N = 0.15 mm, G = 0.3 mm, T = 0.075 mm, D = 0.25 mm. (d) Complex effective refractive index retrieved by S parameter retrieval method. The blue, orange and grey regions represent negative, zero and positive refractive index respectively.

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In this paper, we demonstrate a numerical study of the propagation of an electromagnetic wave through a metamaterial wedge composed of periodically arranged SRRs and metallic wires. Depending on the wavelength of the incident light, the structure exhibits negative, zero, and positive refraction. It is displayed that for all the three types of refractions, the wavefronts initially propagate through the interface along the positive refraction direction, then reorganize, and finally propagate along the effective refracted directions after a delay time required for the establishment. The establishment time for negative or zero refraction is longer than that for positive refraction. Hence, there is no causality breakdown or speed limit violation. Moreover, the phenomena of infinite wavelength and uniform phase in ZIMs as well as phase velocity $v_p$ antiparallel to the group velocity $v_g$ in NIMs are observed.

Simulation and discussion

To present the three types of refractions, we design a wedge-shaped structure as shown in Fig. 1(b), which is periodically stacked on z axis and specifically arranged on xy plane by unit cells to form a wedge with the angle of 45°. Schematic diagram of a unit cell is given in Fig. 1(c), which shows a composite material consisting of 3-layer sandwich structure. The 0.075-mm-thick top layer (a gold SRR) and bottom layer (a gold wire) are deposited on opposite sides of a 0.25-mm-thick dielectric substrate. Detailed parameters of the SRR and the wire can be found in the caption of Fig. 1(c). FR-4 is chosen as the substrate material due to its low cost in practice. From [34], we take the parameter of $\epsilon =4.4$ and loss tangent of 0.02. To investigate the electromagnetic property of the structure, we firstly calculated the effective refractive index. Based on the S parameters retrieval method, the effective n can be calculated [34], as shown in Fig. 1(d). A remarkable dispersion appears in the range from 5 GHz to 20 GHz. Considering light propagating in media, the imaginary part of the effective index is demanded to be small. Thus the spectra that we are interested in are divided into 3 regions: (a) 9.5-11.4 GHz, (b) 11.4-13 GHz, (c) beyond 13 GHz. In the three regions, the real part of refractive index is negative, nearly zero and positive, respectively.

According to Fig. 1(d), three frequencies, 10.6 GHz (n = −1.04), 11.8 GHz (n = 0), and 14.1 GHz (n = 0.65) are chosen to demonstrate the three types of refractions. Three-dimensional finite-difference time-domain methods (Lumerical Solutions) are employed to calculate the propagation of a wave through the wedge made up of unit cells composed of SRRs and wires. To simplify the model, we use perfect electric conductor (PEC) to replace the gold due to its nearly infinite conductivity in GHz region. Because of the periodicity on z axis, periodic boundary condition is applied in this direction. As shown in Fig. 1(b), the incident wave comes from the left, shoots normally onto the first interface of the wedge without refraction, propagates in the wedge, reaches the second interface, and then refraction happens.

The occasion of negative refraction is first considered. As is mentioned before, we choose a Gaussian wave as the incident light at 10.6 GHz, where the effective refractive index is −1.04. Visualization 1 shows the evolution from the incident light just shooting on the first interface to finally establishing the steady propagation, which gives a picture exhibiting the real-time and real-space interaction between light and the structure. From Visualization 1, we select some characteristic moments, as shown in Fig. 2(a) - 2(f). Figure 2(a) shows that the wave just reaches vertex A of the second interface at t = 0.15 ns (0.15 ns after the wave was emitted) when the wave starts to refract at the second interface of the wedge. Figure 2(b) shows that the wave propagates through the center of the wedge with an obvious positive refraction angle at t = 0.43 ns. To present more clearly, the yellow allow is adopted to represent the positively refracted direction. As shown in Fig. 2(c), the wave arrives at vertex B₁ at t = 0.70 ns when the wave propagates through the whole second interface. A pronounced positive refracted direction can be observed in the bottom region closed to vertex B₁, which is the same as Fig. 2(b). Interestingly, the top region near vertex A appears a negative refracted direction (the red arrow) and the middle area a nearly-zero-refracted direction (the purple arrow). Figure 2(d) shows that the wave propagating along the positively refracted direction [the yellow arrow in Fig. 2(c)] gradually turns to the zero-refracted direction [the purple arrow in Fig. 2(d)] at t = 0.79 ns. At the same time, the wave along the zero-refracted direction [the purple arrow in Fig. 2(c)] turns to the intermediate negatively refracted direction [the red arrow in Fig. 2(d)]. In Figs. 2(e) and 2(f), the wave along the negatively refracted direction is more and more pronounced, since the waves propagating along other two directions have turned to the negatively refracted direction. Finally, the wave propagates with a negative refraction angle of −45° [shown in Fig. 2(f)], which is expected from Snell’s law.

 

Fig. 2 (Visualization 1) Distribution of the electric field ($\ln (|E|)$) overlapped with the Poynting vector for the model. Effective refractive index of the wedge-shaped structure is n = −1.04 at 10.6 GHz. (a) t = 0.15 ns. (b) t = 0.43 ns. (c) t = 0.70 ns. (d) t = 0.79 ns. (e) t = 1.25 ns. (f) t = 1.50 ns. The wedge structure is emphasized by white solid lines, and the white dashed line is the normal of the second interface of the wedge. The yellow, red, and purple arrows represent the positively, negatively and zero-refracted direction, respectively.

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From the transient establishment process (Visualization 1), it can be verified that negative refraction does not exceed the speed limit c. More specifically, we find that the negatively refracted wavefront has already appeared in Fig. 2(c), and becomes more and more pronounced in Figs. 2(d)-2(f). Obviously, the upper part of the wave near vertex A establishes the negatively refracted wavefront primarily, due to a shorter optical path between the two interfaces of the wedge. However, we notice a transient establishment of the negative refraction [as seen from Figs. 2(a)-2(e)], which precedes the steady propagation along the negatively refracted direction [as shown in Fig. 2(f)]. It means that the B-B₁-B' process [Fig. 2(f)] does not finish at an infinite speed [as described in Fig. 1(a)]. On the contrary, the whole process of establishment takes about 1.35 ns to reorganize a negatively refracted wavefront and propagate eventually along negatively refracted direction. It is longer than the time difference (0.55 ns) of arrival at the second interface between the vertex A [in Fig. 2(a)] and B₁ [in Fig. 2(c)] of the wave. This phenomenon solves the manifest paradox of the superluminal propagation of point B.

The same way as negative refraction is used to study zero and positive refraction. 11.8 GHz with n = 0 is chosen to simulate zero refraction. As Figs. 3(a)-3(c) shows the wave reaches the vertex A, center and vertex B₁ of the second interface at t = 0.17 ns, t = 0.45 ns and t = 0.72 ns respectively. Finally, Fig. 3(d) shows that the wavefront propagates along the normal at t = 0.91 ns, which is also expected from Snell’s law. As shown in Figs. 3(b) and 3(c), two wavefronts appear in the refracted wave, moving toward zero-refracted direction and positively refracted direction. Similar to negative refraction, the positively refracted direction [the yellow arrow in Fig. 3(b)] gradually turns to the zero-refracted direction [the purple arrow in Fig. 3(c)]. The transient establishment of the zero refraction [from Figs. 3(a)-3(c)] precedes the steady propagation along the zero-refracted direction [in Fig. 3(d)]. This means that the point B does not move instantaneously to B'. But the whole process of establishment takes about 0.74 ns [from Figs. 3(a)-3(d)] to reorganize a zero-refracted wavefront and propagate eventually along zero-refracted direction. It is longer than the time difference (0.55 ns) of arrival at the second interface between the vertex A [in Fig. 3(a)] and B' [in Fig. 3(c)] of the wave. Hence, there is no violation of the speed limit either.

 

Fig. 3 (Visualization 2) Distribution of the electric field (magnitude) overlapped with the Poynting vector for the model. Effective refractive index of the wedge-shaped structure is n = 0 at 11.8 GHz. (a) t = 0.17 ns. (b) t = 0.45 ns. (c) t = 0.72 ns. (d) t = 0.91 ns. The wedge structure is emphasized by white solid lines, and the white dashed line is the normal of the second interface of the wedge. The yellow and purple arrows represent the positively and zero-refracted direction, respectively.

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14.1 GHz with n = 0.65 is selected to exhibit positive refraction. Figures 4(a)-4(c) shows the wave reaches the vertex A, center and vertex B₁ of second interface at t = 0.17 ns, t = 0.43 ns and t = 0.71ns respectively. Finally, it propagates along the effective refracted direction (27°), which obeys the Snell’s law as shown in Fig. 4(d) at t = 0.80 ns. Two positively refracted wavefronts appear in the refracted wave propogating along two positively refracted directions in Figs. 4(b) and 4(c). The positively refracted direction that deviates larger from the normal [the yellow arrow located in the center of Fig. 4(b)] gradually turns to the other direction [27° the yellow arrow in the center of Fig. 4(c)]. According to the simulation, the establishment of positive refraction needs 0.63ns [from Figs. 4(a)-4(d)].

 

Fig. 4 (Visualization 3) Distribution of the electric field (magnitude) overlapped with the Poynting vector for the model. Effective refractive index of the wedge-shaped structure is n = 0.65 at 14.1 GHz. (a) t = 0.17 ns. (b) t = 0.43 ns. (c) t = 0.71 ns. and (d) t = 0.80 ns. The wedge structure is emphasized by the white solid lines, and the white dashed line is the normal of the second interface of the wedge. The yellow arrows represent the positively refracted direction.

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In order to interpret the establishment time of different effective refractive indexes, two more frequencies (10.9 GHz and 15.2 GHz) with n = −0.66 and n = 0.92 have been calculated. The five results are listed in Table 1. For all the refraction processes, the wavefronts initially propagate through the second interface along the positive refraction direction, then reorganize, and finally propagate along the corresponding refracted directions. Clearly, the establishment time is longer when the effective refractive index is negative. Besides, for negative refraction, we can observed from Visualization 1 that the previous several wavefronts of the wave propagate through the first interface of the wedge with a positive $v_p$ first and propagate through the second interface with a positively refracted direction. The phase velocity of the subsequent wavefronts gradually turn to the negative $v_p$ while the refracted wave gradually turns to negatively refracted direction. Meanwhile, the energy moves forward all the time, which mean a positive $v_g$. Likewise with zero refraction, the previous several wavefronts of the wave propagate through the first interface with a positive $v_p$ first and propagates through the second interface with a positively refracted direction. The phase of the subsequent wavefronts gradually become uniform and the refracted wave turns to zero-refracted direction (Visualization 2).

Table 1. Establishment time of wavefronts for different frequencies.

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Conclusion

In this paper, we investigate the propagation of a wave through a wedge metamaterial composed of periodic SRRs and metallic wires. The results quantitatively demonstrate the transient establishment of wavefronts for negative, zero and positive refraction. The establishments of negative, zero and positive refraction need around 1.35 ns, 0.74 ns and 0.63 ns respectively to reorganize the wavefronts to the final directions. The establishment time of negative or zero refraction is 0.72 ns or 0.11 ns longer than that for positive refraction. Hence, these transient processes guarantee that negative and zero refraction occur without breaking causality or violating special theory of relativity. For all three refraction processes, transient establishment precede the steady propagation, and the upper parts of incident wavefronts first form the refracted wavefronts due to the shorter optical path between the two interfaces of the wedge. Infinite wavelength and uniform phase are found inside the ZIM and the phase velocity $v_p$ turns out to be antiparallel to the group velocity $v_g$ in the NIM. The results provide some new insights on the interaction between electromagnetic waves and metamaterials.