1. INTRODUCTION

Negative refraction from periodic structures has been observed in both physical experiments [1–3] and numerical computations [4,5]. Early on, it was shown that a composite medium of wires and split metal loops would present an effective negative index of refraction to waves of the appropriate polarization [6]. Shelby, Smith, and Schultz subsequently performed the original experiment demonstrating negative refraction in 2000 [1], when they measured the refracted beam from a triangular prism constructed of unit cells arranged into a staircased prism.

Many researchers compute effective medium properties from the induced average polarization and magnetization responses of the unit cells [7–9]. In particular, Liu and Alu have carried out calculations for the periodic medium composed of densely spaced magneto-dielectric spheres [4]. They concluded from their analysis that a dipolar-effective medium treatment predicts negative indices of refraction that are consistent with full-wave electromagnetic computations.

In contrast to these approaches, we develop a predictive model for negative refraction in triangular prisms based on finite array theory that does not depend on the prior determination of an appropriate effective medium description. Our model emphasizes the importance of a single property of the underlying unit cell of the periodic medium: the excess phase delay imparted by the unit cell on a wave during transmission across the cell, compared to the phase delay across an equal distance of free space. Starting from a knowledge of the excess phase delay and the geometry of the prism, we develop an equivalent linear array of aperture radiators. The main transmitted beam of the equivalent aperture array corresponds to the main beam transmitted through the prism.

Using our model, we revisit the staircased prism made of periodically arranged spherical magneto-dielectric unit cells as proposed by Liu and Alu [4]. We compare theoretical predictions of the beam transmission angle through the prism to those simulated in full-wave electromagnetic software. The agreement is excellent over the range of frequencies where the prism allows for beam transmission without appreciable reflections. The success of this model in predicting the occurrence of negative refraction suggests that, in fact, one need only select a unit cell with the correct phase delay to produce negative refraction.

2. THEORY

The behavior of a negatively refracting, triangular metamaterial prism can be explained by combining array theory with a suitable field equivalence theorem. We consider a triangular prism with interior wedge angle $\alpha$ constructed with cubic unit cells with edge length $d$. The prism is not a perfect triangle, as its hypotenuse is approximated as a staircase that proceeds in steps according to the unit cell edge lengths. If we restrict ourselves to steps one cell in height and $R$ cells wide, then we can achieve interior wedge angles

In our examples, we set $R=4$, giving an interior wedge angle $\alpha \approx 14°$.

We presume an electromagnetic plane wave is incident upon the long, flat edge of the triangular prism, pictured in Fig. 1, and we measure the beam transmission angle $\varphi$ counterclockwise from the positive $x$ axis. The prism is negatively refracting if $\varphi >\alpha +\pi /2$. We define a refraction angle $\theta =\frac{\pi }{2}+\alpha -\varphi$, measured from the normal line to the hypotenuse. Negative refraction is characterized by beam transmission at angles $\theta <0$.

 

Fig. 1. Prism geometry.

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Next, we draw the surface $S$ over the prism and divide this surface into N strips of equal width, according to Fig. 2.

 

Fig. 2. Array theory geometry.

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We imagine each column as a transmission line feeding an aperture radiator situated upon its terminal edge. By invoking a field equivalence principle [10], we may replace the wedge with equivalent magnetic surface currents lying on $S$ radiating into free space. We assume that the currents on all terminal edges are identical except for their phases. This allows us to model the metamaterial prism as a scanning aperture array and to compute the beam transmission angle using the element pattern and array factor.

In our formulation, the key parameter of the metamaterial unit cell is the excess phase delay it delivers at each frequency, compared to an equal distance of free space. For instance, suppose the unit cell was a solid block of magneto-dielectric material with relative permeability ${\mu }_r$ and relative permittivity ${ϵ}_r$. The excess phase delay of the unit cell in this case would be

The number of unit cells encountered along each column is one more than the previous column. Thus, the transmission lines are configured to deliver a progressive phase delay to the radiating elements in the aperture array. We suppose that the transmitted electric field through the prism, along the surface $S$, is

Invoking the perfect electric conductor (PEC) exterior equivalence [10], we place magnetic surface currents on $S$ equal to

The transmitted beam pattern, according to Eqs. (13)–(15), is a function of both the element pattern and the array factor. While the effect of the element pattern needs to be incorporated for an accurate determination of the actual angle of refraction, its contribution is secondary in importance to that of the array factor. Typically, array theory interprets Eq. (16) solved for $m=0$ as a main beam while solutions for $m=\pm 1,\pm 2,\dots$ are interpreted as grating lobes. For progressive phase delays less than $\pi$, grating lobe solutions are dependent on element spacing exceeding ${\lambda }_0/2$. However, with $\psi$ exceeding $\pi$, and spacing less than ${\lambda }_0/2$, it is entirely possible to obtain solutions for $m=\pm 1,\pm 2,\dots$ even when the $m=0$ case cannot be solved. Such solutions may exist in angular directions described as negatively refracted beams.

For an approximate estimation of the frequency interval for the occurrence of negative refraction, we can temporarily disregard the effect of the element pattern. Accordingly, under this approximation, from Eq. (16), for negative refraction, $\varphi >\alpha +\frac{\pi }{2}>\frac{\pi }{2}$; therefore $\cos (\varphi )<0$, and the first occurrence of negative refraction will be when $m=-1$. Setting $m=-1$ and solving Eq. (16) for $\varphi$, we obtain

Substituting Eq. (17) into the inequality $\alpha +\frac{\pi }{2}<\varphi <\pi$ and solving the inequality for $\psi$ yields

In the above inequality, both ${\lambda }_0$ and $\psi$ are functions of frequency. The inequality in Eq. (18) can be utilized to estimate the frequency interval for negative refraction before engaging in detailed calculations. The estimate for the actual angle of refraction is obtained as the angle $\varphi$ where the main beam within the transmitted pattern reaches a maximum, as calculated using Eq. (13)–(15).

3. NUMERICAL RESULTS

We performed numerical analysis of a 14° prism of magneto-dielectric spheres to test the theory proposed in Section 2. The unit cell is a cube with edge length $d$ with a sphere of radius $r$ in the center. The unit cell and prism design are based on those proposed by Liu and Alu [4]. The geometry of the unit cell is shown in Fig. 3(A), and the prism is shown in Fig. 3(B). The sphere is made of a hypothetical material with an index of refraction $n=\sqrt{{ϵ}_r{\mu }_r}=12.3$ as proposed in [4], and the region of the cell outside of the sphere is free space. We examine the prism over the frequency range $0.514<k_{0}d<0.807$.

 

Fig. 3. Geometry of unit cell and prism. (A) Cubic unit cell with edge length $d$ containing a magneto-dielectric sphere of radius $r=0.45d$. (B) 14° prism of magneto-dielectric spheres with waveguide source for simulation. Both panels show the coordinate axes of their respective geometries.

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We use XFdtd [12], a commercial implementation of the finite-difference time-domain method [13,14], to perform our numerical experiments. The time-domain approach allows us to examine how negative refraction develops with time in addition to determining the steady-state refraction angles. The mesh grid is adaptive, with a mesh size of $0.045d$ inside and around the spheres and a coarser mesh grid with edge lengths of $0.57d$ in the regions away from the prism.

A. Unit-Cell Response of Magneto-Dielectric Sphere

We compute the response of the unit cell by simulating it in a transverse-electromagnetic (TEM) waveguide with perfect magnetic conductor (PMC) boundaries on the $\pm x$ faces and perfect electric conductor (PEC) boundaries on the $\pm z$ faces. We placed TEM waveguide ports on the $\pm y$ faces of the unit cell, as shown in Fig. 3(A). The port on the $-y$ face transmits a Gaussian pulse, while the port on the $+y$ face serves as a sensor to measure the time-domain received signal. XFdtd uses the discrete Fourier transform to convert time-domain signal data to frequency-domain scattering parameters. We compute the excess phase delay of the unit cell by taking the difference between the unwrapped phase of the S21 scattering parameter and the phase delay of a cell filled with free space of the same size. The excess phase delay is shown in Fig. 4. From Eq. (18), our wedge with $\alpha \approx 14°$ and unit cell sizes on the order of $d/{\lambda }_0\approx 1/9$ will refract negatively when the excess phase delay lies approximately within the range $200°<\psi <320°$. Using these numbers, Fig. 4 predicts that negative refraction will occur within the frequency range of $0.679<k_{0}d<0.761$.

 

Fig. 4. Excess phase delay of unit cell with magneto-dielectric sphere versus frequency and expected negative refraction region (demarcated by dotted lines).

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B. Prism of Magneto-Dielectric Spheres

We model the 14° prism as a single layer of spheres with periodic boundary conditions on the $\pm z$ boundaries of the simulation to create the effect of an infinitely long prism. We use perfectly matched layer absorbing boundary conditions on the other four sides of the simulation space. Our source is a TEM waveguide port polarized with its electric field aligned with the $z$ axis (and thereby the axis of the prism). The port is 24 unit cells wide and 1 unit cell high (the full height of the simulation space). We transmit a ramped-sinusoid waveform through the port and record the time-domain electric fields with a planar sensor on the $x–y$ plane. Figure 3(B) shows the prism and waveguide ports along with the coordinate axes for the simulations.

Steady-state, instantaneous electric-field magnitudes are shown in Fig. 5. Figure 5(A) shows the electric field for $k_{0}d=\mathrm{0.514.}$ At this frequency, the prism is positively refracting. Figures 5(B) and 5(C) show the negatively refracted beams at $k_{0}d=0.660$ and $k_{0}d=0.734$, respectively. At $k_{0}d=0.807$, the prism is again positively refracting, as is shown in Fig. 5(D).

 

Fig. 5. Instantaneous electric fields refracted through 14° prism of magneto-dielectric spheres. All panels are scaled relative to the peak amplitude in that panel. (A) $k_{0}d=0.514$ with peak amplitude of 5.1 kV/m. (B) $k_{0}d=0.660$ with peak amplitude of 13.2 kV/m. (C) $k_{0}d=0.734$ with peak amplitude of 10.5 kV/m. (D) $k_{0}d=0.807$ with peak amplitude of 6.9 kV/m.

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Figure 6 compares the transmitted beam angle computed in XFdtd with the array-theoretic predictions. The results from XFdtd match the array-theoretic predictions within 5° over the range of $k_{0}d$ studied. The XFdtd results show a transition from positive to negative refraction between $k_{0}d=0.624$ and 0.660, as predicted by the array theory model.

 

Fig. 6. Beam transmission angle through 14° prism of magneto-dielectric spheres: Array theory versus xFDTD simulation.

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The theoretical calculation of the refracted angle is taken as the angle of peak transmitted electric field magnitude, as determined by the product of the element pattern in Eq. (14) and the array factor in Eq. (15). The progressive phase shift used in the array factor is shown in Fig. 4. As the excess phase delay through each unit cell increases with frequency, $\theta$ increases until it reaches nearly 90°. At that point, $\theta$ is ill-defined as the beam is scanned through the nulls of the element pattern at $\varphi =0°$ and $\varphi =180°$. The array-theoretic model is somewhat unreliable over the range $0.660<k_{0}d<0.679$, probably as a result of the rapid change of the beam with frequency. It is doubtful that the negatively refracting wedge could be physically realized over this frequency range, since the array factor is scanning close to the null directions of the element pattern.

The array-theoretic model accounts for interactions between cells in the prism by utilizing a phase delay for each unit cell calculated in the presence of a laterally infinite array of cells. Our theoretical model yields accurate estimates for the transmitted beam angle over a wide range of frequencies.

4. CONCLUDING REMARKS

While both physical experiments [1–3] and numerical computations [4,5] have demonstrated the existence of negatively refracted beams, we have shown that negative refraction can be explained without assigning an effective index of refraction. Array theory provides an alternative explanation of negative refraction and gives accurate predictions for the case studied.

Array-theoretic predictions of the frequency range for negative refraction, according to Eq. (17), were seen to depend on the interior wedge angle $\alpha$. The array-theoretic model accounts for dependencies of the refractive index of the prism upon the prism geometry itself. In particular, the model connects negative refraction with the staircasing of the prism, through the progressive phase shift in the equivalent aperture array. Furthermore, the array model takes into consideration the lateral extent of the impinging beam upon the wedge.

The model also provides answers to objections raised by Valanju et al. [15] and Munk [16]. Valanju et al. argue that negative refraction is impossible from negative index media using arguments based on causality and dispersion of such effective media. However, in the XFdtd simulations, a number of cycles pass before the formation of the beam. Indeed, the in-phase addition needed to form the negatively refracted beam is understood to be an addition of wavefronts from each individual radiator delayed by $2\pi$ (or integer multiples of $2\pi$).

Munk, on the other hand, considers that negative refraction might result from the formation of a grating lobe in the far field when a single sheet of unit cells is excited by a uniform plane wave. However, Munk rejects this interpretation because the lateral unit cell spacing for a valid effective medium description is at least smaller than a half-wavelength, precluding grating lobe formation. By contrast, in our model, each radiating aperture is preceded by a column of unit cells delivering a progressive phase shift independent of the lateral spacing between unit cells. This admits the possibility of a grating lobe, and by extension, of negatively refracted beams [17]. The negatively refracted beam is identified as a grating lobe, in accordance with setting $m=-1$ in Eq. (16).

In this paper, we considered negative refraction through a triangular prism comprising magneto-dielectric sphere unit cells. On the basis of the effective medium properties ascribed to these unit cells, other interesting behaviors have been predicted for different geometrical arrangements, such as super-lensing through a slab [18,19]. It would be interesting to apply the array-theoretic perspective, for whatever insights it may provide into the underlying mechanisms.