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Superscatterer is an intriguing electromagnetic device, which can enhance the wave scattering of a given object with an arbitrary magnification factor in principle. However, observing the transient buildup of a superscatterer in numerical time domain still has not been investigated yet. In this paper, by using the dispersive finite difference time domain method, the transient response of a dispersive superscatterer created with monotonic optical transformation function is studied. We find that the time delay grows dramatically when the magnification factor increases. In addition, we notice an interesting phenomenon that, placing a scattering body with more complicated structure leads to longer time delays. These findings are very useful to reveal the physics behind the superscatterer.
© 2017 Optical Society of America
1. Introduction
Superscatterer [1–3] is a kind of electromagnetic device that can magnify the scattering cross section of a given object significantly and has potential applications in the fields of imaging, spectroscopy, biomedicine and photovoltaics [4, 5]. This concept was first proposed in the framework of transformation optics [6–13], and realized by coating an anisotropic and inhomogeneous electromagnetic shell on a perfect electric conductor (PEC) cylinder. To loose the parameter requirement, another method proposed by Ruan and Fan [14, 15] only requires conventional isotropic plasmonic and dielectric materials. They designed the subwavelength superscatterer by engineering an overlap of resonances of different plasmonic modes. Moreover, Li et al. further extended to the deep-subwavelength scale using graphene monolayers with the advantage of active tunability [16–18]. Here we focus on the first method based on the concept of transformation optics.
Inspired by the superscatterer, novel devices with different functionalities and types have been designed, such as a frequency-selective superabsorber metamaterial using an absorbing core material coated with an isotropic double negative shell [19], high-directivity emission with a small antenna aperture [20], and an elliptical electromagnetic superscatterer without rotational symmetry [21]. However, these pioneering works mainly focus on single frequency electromagnetic waves, which are performed in frequency domain based on commercial softwares. So the effects of the dispersion are ignored. Meanwhile, very little attention has been paid to the time domain analysis of the superscatterer. Allowing for the transient illumination, wide spectrum and dispersion [22], studies in time domain [23–29] become more significant because it can fully exploit interesting phenomena during dynamic propagation process that cannot be revealed in frequency domain, such as time delay. While the time delays of the superlens were studied by Gomez Santos [30] and Wee et al. [31], respecticely, the solutions obtained mainly focus on theoretical analysis. A numerical demonstration of the transient response of the superscatterer, which will be very important to reveal the physics behind the phenomena and will also be a nice complement to the previous theoretical work, still has not been investigated.
In this paper, by using the dispersive finite difference time domain (FDTD) method under a cylindrical coordinate, the dynamic processes of a dispersive superscatterer created with monotonic optical transformation function are exhibited. We observe the variations of time delay when increasing the magnification factor. In addition, instead of the cylindrical scatterer, we use an electric fan scatterer to confirm whether time delay is in relation with the structure of scattering body itself. We show that the time to reach steady state will be longer when the structure of the scatterer becomes more complex.
2. Dispersive FDTD model
The schematic of an infinitely long cylindrical superscatterer with inner radius R1 and outer radius R2 is shown in Fig. 1(a). The background is free space surrounded by perfectly matched layers [32]. The plane wave is normally incident from left to right after passing through the dispersive superscatterer. In Fig. 1(a), a big cylindrical region r ≤ R3 is compressed to a small one r ≤ R1, and the region within R2 < r < R3 is folded into the complementary region R1 < r < R2. Therefore, the inner cylinder is magnified to the large cylinder (the dashed line) [1]. This can be achieved by a simple transformation function r′ = f(r), ϕ′ = ϕ, z′ = z, where,
Here we define the magnification factor as m = f(R1)/R2, where f(R1) = R3 and R3 > R2. The schematic figure of f(r) is shown in Fig. 1(b), where the horizontal axis represents physical space and the vertical axis represents virtual space. It can be seen that the outer circle with radius R2 remains unchanged under the transformation while the inner circle with radius R1 is magnified to that with radius R3. Therefore, in this region r > R3, the scattering field is exactly equivalent to that induced by a large cylinder with radius R3. Using coordinate transformation, the anisotropic permittivity ∊̿ = ∊rr̂r̂ + ∊ϕϕ̂ϕ̂ + ∊zẑẑ and permeability μ̿ = μrr̂r̂ + μϕϕ̂ϕ̂ + μzẑẑ within r < R2 are obtained: diag[∊r/∊0, ∊ϕ/∊0, ∊z/∊0] = diag[μr/μ0, μϕ/μ0, μz/μ0] = diag[f(r)/rf′(r), rf′(r)/f(r), f(r) f′(r)/r]. So the constitutive parameters in the region r ≤ R1 are all positive while those in the annulus R1 < r < R2 are negative, as shown in Fig. 1(c).
Fig. 1 (a) The setup of the superscatterer system. An infinitely long cylindrical superscatterer with inner radius R1 and outer radius R2 is placed at the center. The inner cylinder is the scattering body. The blue region is a complementary medium shell. R3 is the radius of the magnified cylinder (dashed line). (b) The schematic figure of the monotonic transformation function, where we can tune the magnification factor m = R3/R2 by adjusting R3. (c) The permittivity and permeability components for the superscatterer. Within the range R1 < r < R2, these three components are all smaller than zero while greater than zero in other range.
Our study focuses on Hz-polarized mode with magnetic field perpendicular to the rϕ-plane of the setup, where only the permittivities ∊r, ∊ϕ and the permeability μz are involved. For each Yee’s grid in the dispersive FDTD method, the material parameters ∊r, ∊ϕ and μz are mapped with dispersive models when their values are smaller than the ones in the free space. So, in the complementary region of this superscatterer, dispersive models are employed because of the double negative permittivity and permeability. Drude model [33] is applied to ∊r, ∊ϕ and Lorentz model [34] is adopted for μz, as follows:
where γ is the dissipation factor, ωpr, ωpϕ and ωpz are plasma frequencies, and ω0 is the resonant frequency. We adopt the recursive convolution method to deal with frequency-dependent material parameters and obtain iterative equations [25]. Theoretically, we assume the materials are lossless by setting the imaginary parts of Eqs. (2)–(4) to be zero, and ω0 is set to be equal to ωpz [23,25,35]. In practice, for each Yee’s grid, we regulate the plasma frequencies to make the real part of Eqs. (2)–(4) equal to the specified values ∊r, ∊ϕ and μz calculated from Eq. (1) at the superscattering frequency.3. Numerical results and discussion
In the two-dimensional FDTD simulation, we set R1 = 5 mm, R2 = 12 mm and R3 = 16.8 mm as an example. A Hz-polarized time-harmonic excitation is characterized as J̄m (t) = ẑcos(2πf0t) with f0 = 32.2 GHz and located at x = −64.8 mm. Here we let the superscattering frequency equal to f0. The inner cylindrical region r ≤ R1 is a PEC.
Figure 2 shows the snapshots of the longitudinal magnetic field distribution in two cases: Figs. 2(a)–2(e) are induced by the superscatterer (R1 = 5 mm; R2 = 12 mm; R3 = 16.8 mm) and Fig. 2(f) is induced by the equivalent cylindrical PEC with radius R3. In Figs. 2(a)–2(d), the superscattering effect is built up gradually, and finally gets to the steady state in Fig. 2(e). Comparing the magnetic field distribution in the region r > R3 of Figs. 2(e) and 2(f), we can find they are almost the same, which indicates that the inner cylindrical PEC is magnified to the large cylinder PEC by coating the complementary medium shell. During the dynamic process, we monitor the transmitted Hz field at many typical points such as A(16.8, 21.6) mm, A′(20.8, 21.6) mm, B(16.8, 14.4) mm, B′(20.8, 14.4) mm, C(16.8, 7.2) mm and C′(20.8, 7.2) mm in Fig. 3(a). Temporal evolutions of magnetic field at points A, B and C are exhibited in Figs. 3(b)–3(d) for examples. The red solid line and the blue dashed line represent the time evolution of Hz in the superscatterer and the equivalent cylindrical PEC, respectively. When the magnetic amplitude does not change or only shifts in a certain error range (Δ < 1%) for more than ten periods, we can consider the field reaches the steady state. Until all positions reach the steady state, the superscatterer can be regarded as steady. Here we define time delay, or steady state time as the time span between the start moment when the signal first arrives at the observation point and the end moment when it reaches steady state. We find that the time delay is almost the same when the measured points are located at the same y-coordinate. In the current case, the time delay measured at point B(16.8, 14.4) mm is the longest as shown in Fig. 3(e). Starting from t = 271 ps to t = 1712 ps, time delay is 1441 ps, corresponding to 46.4 periods. Physically, under transient illumination, the complementary medium coating will concentrate the incoming wave into the internal region before reaching the steady state. During this period, the whole system will store some energy from the incident wave. The stored waves will be scattered from the inner PEC core, and with the complementary medium coating, they will constructively interfere with the incoming wave. Gradually, the scattering enhancement phenomena are formed. This accounts for why such a superscatterer phenomenon requires a period of time delay to store energy, then to be established.
Fig. 2 Time-evolution of the longitudinal magnetic field Hz at different moments (a) t = 222.60 ps, (b) t = 364.35 ps, (c) t = 501.06 ps, (d) t = 650.58 ps, (e) t = 2510.13 ps. The Hz-polarized plane wave is incident from x = −64.8 mm to the superscatterer (R1 = 5 mm; R2 = 12 mm; R3 = 16.8 mm). (f) The total magnetic field induced by the equivalent large cylindrical PEC with radius 16.8 mm at t = 2510.13 ps.
Fig. 3 (a) Propagation of Hz-polarized plane wave through a superscatterer (R1 = 5 mm; R2 = 12 mm; R3 = 16.8 mm) with an input plane set up at x0 = −64.8 mm. Many typical points such as A(16.8, 21.6) mm, A′(20.8, 21.6) mm, B(16.8, 14.4) mm, B′ (20.8, 14.4) mm, C(16.8, 7.2) mm and C′ (20.8, 7.2) mm are monitored. Time evolution of magnetic field Hz measured at (b) point A, (c) point B and (d) point C in the cases of the superscatterer and the equivalent cylindrical PEC. (e) Time evolution of the amplitude of Hz measured at point B for superscatterer. Time delay starts from t = 271 ps to t = 1712 ps.
In the following, we continue to observe time delays when modifying the magnification factor m = R3/R2 by tuning R3. As depicted in Fig. 4, the curve of the time delay goes up steeply when the magnification factor increases from 1.2 to 2. Theoretically, a lossless super lens requires infinite time to reach steady state [30, 31]. The reason why our results show a finite time delay, is that the cut-off of the wavenumber is determined by the finite dimension of the Yee’s grid spacing used in the dispersive FDTD. Another reason is due to the definition of steady state, i.e., we will deem the system reaches steady state when the magnetic amplitudes do not change or only shift in a certain error range (Δ < 1%) for more than ten periods. Theoretically, for a real ideal steady state, Δ should be an infinitesimal value, but practically we can not achieve this in a numerical method. Moreover, in Wee et al.’ theoretical prediction [31], for the same magnification factor 2, it would take roughly 60 ps to reach steady state while it is 4300 ps in our case. This is because Wee et al. consider the magnified image reaches the steady state when the magnetic amplitudes are in the error range (Δ < 8%) approximately. If we use the same standard, the numerical simulation shows about 1200 ps time delay, which is still one order than Wee et al.’ theoretical prediction. This difference is mainly caused by the losses σ = 1.8 × 10−3 used in the cylindrical perfect lens by Wee et al. As our numerical simulation only consider lossless case, it is very reasonable that longer time delay is required to reach steady state.
Fig. 4 The time delay characteristic when the magnification factor m = R3/R2 ranges from 1.2 to 2. The curve goes up steeply when the magnification factor increases. The two insets above the curve are the corresponding steady magnetic field distributions at t = 2142.3 ps and t = 4015.8 ps, respectively.
In order to study how the geometry of the scattering body affect the time delay, we use a completely different scattering body shaped like an electric fan shown in the right-bottom inset of Fig. 5(d). The electric fan made of PEC owns four leaves whose angle θ = 32° and a center disk. For the superscatterer, the inner radius and the outer radius are consistent with the aforesaid ones, i.e., R1 = 5 mm and R2 = 12 mm, and the radius of the center disk always remains 2 mm. More specifically, R3 is the sole changing parameter, which leads to different parameter constitution inside the superscatterer. For different magnification factor, the overall geometric size is magnified to an equivalent large PEC electric fan with radius R3 while the whole shape remains the same. Taking R3 = 16.8 mm as an example, Figs. 5(a) and 5(b) show the Hz field distributions at the moments of t = 622.65 ps and t = 4200 ps, respectively. Figure 5(c) shows the magnetic field scattered by an equivalent large PEC electric fan with radius 16.8 mm at t = 4200 ps. In agreement with the aforementioned analysis, the patterns of the magnetic field within region r > R3 in Figs. 5(b) and 5(c) are nearly identical. The time delay for different magnification factor is shown in Fig. 5(d). Compared with the cylindrical scattering body in Fig. 4, the steady state time becomes longer for the same magnification factor. One possible explanation is that the electric fan scatterer owns many edges and corners, which brings about more evanescent waves to accumulate in order to reach steady state. This indicates that the steady state time is related with the structure of the scattering body.
Fig. 5 Distribution of the magnetic field when the Hz-polarized plane wave is incident onto the scatterer shaped like an electric fan coated by a complementary media at (a) t = 622.65 ps, (b) t = 4200 ps. (c) The total magnetic field scattered by the equivalent large PEC fan at the same moment with (b). (d) The time delay characteristic when the magnification factor m = R3/R2 ranges from 1.2 to 2. The two insets above the curve are the steady snapshots at t = 5145 ps and t = 8400 ps, respectively. The right-bottom inset shows the schematic of the electric fan scatterer.
4. Conclusion
In conclusion, the transient properties of the dispersive superscatterer are investigated, and time delays are revealed using the dispersive FDTD method. Our study shows that the time to reach steady state grows dramatically when the magnification factor of the superscatterer increases. Moreover, we study the relation between time delay and the structure of the scattering body. We notice a longer time delay because of the complex structure of the electric fan. This study gives deep insight into the superscatterer based on transformation optics in time domain, and will have profound effects on superscatterer design. The transient analysis method proposed in this paper can be also applied in other novel electromagnetic devices [36–38].
Funding
National Natural Science Foundation of China (NSFC) (61625502, 61574127 and 61601408); the ZJNSF (LY17F010008); the Postdoctoral Science Foundation of China (2015M581930); the Top-Notch Young Talents Program of China; the Fundamental Research Funds for the Central Universities; the Innovation Joint Research Center for Cyber-Physical-Society System.
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