Dispersive cylindrical cloaks under nonmonochromatic illumination

Transformation-based cylindrical cloaks and concentrators are illuminated with nonmonochromatic waves and unusual effects are observed with interesting potential applications. The transient responses of the devices are studied numerically with the finite-difference time-domain method and the results are verified with analytical formulas. We compute the effective bandwidth of several cloaking schemes as well as the effect of losses on the performance of the structures. We also find that narrowband behavior, frequency shift effects, time delays, and spatial disturbances of the incoming waves are dominant due to the inherently dispersive nature of the devices. These effects are important and should be taken into account when designing metamaterial-based devices.

Figure 1

(Color online) (a) FDTD computational domain of the cylindrical cloaking structure for the case of nonmonochromatic plane wave illumination. The fields are computed using the total-field scattered-field (TF-SF) technique (Ref. [33]). (b) Comparison between Drude and Lorentz dispersion models, when used to map the ${\widehat{\epsilon }}_r$ parameter at the point $r=R_1$. The bandwidth of a typical incident pulse is also plotted in the same graph to illustrate the dispersion along its spectrum.

Figure 2

(Color online) (a) The transmission amplitude of the ideal, matched reduced and practical reduced cylindrical cloak. The transmission of a bare PEC cylinder is also plotted to calculate the cloak’s effective bandwidth. (b) The transmission amplitude of the ideal cylindrical cloak when different losses are introduced.

Figure 3

(Color online) Blueshift effect observed in the normalized frequency spectra of transmitted Gaussian narrowband pulses through the ideal cylindrical cloak for two averaging line segments $L_1$ and $L_2$ of Fig. 1a. The line segments are on the same position on the $y$ axis, $1.5\lambda$ away from the cloak’s outer shell, but have different lengths along the $x$ axis $\left(L_1=R_1,L_2=4.25R_2\right)$. The effect is stronger for the line segment $L_1$, near the center of the cloaking structure.

Figure 4

(Color online) Frequency spectra of six transmitted narrowband pulses, centered at different frequencies, after impinging on the ideal cloak. The cloak’s design frequency is 2 GHz. The numbers next to each plotted curve indicate the incident pulse’s central frequency, ranging from 1.7 to 2.3 GHz. (a) Fields averaged over the short segment $L_1$ immediately after the cloak’s boundary. (b) Fields averaged over the long segment $L_2$ immediately after the cloak’s boundary. (c) Fields averaged over the short segment $L_1$ $1.5\lambda$ away from the cloak’s boundary. (d) Fields averaged over the long segment $L_2$ $1.5\lambda$ away from the cloak’s boundary.

Figure 5

(Color online) (a) Normalized penetration depths for ideal cylindrical cloaks with different thicknesses, computed analytically. (b) Regions formed inside the ideal cylindrical cloak, when it operates at $f<f_0$ frequencies. (c) Normalized penetration depths for matched reduced cylindrical cloaks with different thicknesses, computed analytically. The numerically estimated penetration depths are also shown for a cloak with thickness $2\lambda /3$. The error bars indicate numerical uncertainty on the field cutoff radius. (d) Regions formed inside the matched reduced cylindrical cloak, when it operates at $f<f_0$ frequencies.

Figure 6

(Color online) Frequency spectra of six transmitted narrowband pulses, centered at different frequencies, after impinging on the matched reduced cloak. The cloak’s design frequency is 2 GHz. The numbers next to each plotted curve indicate the incident pulse’s central frequency, ranging from 1.7 to 2.3 GHz. The fields are averaged over two different line segments $L_1$ (a) and $L_2$ (b), as shown in Fig. 1a, which are perpendicular to the direction of propagation. The line segments are positioned at a distance $1.5\lambda$ away from the cloak’s outer shell.

Figure 7

(Color online) Real part of the magnetic field amplitude distributions $\left(H_z\right)$ when plane waves of different frequencies are impinging on the ideal [(a)–(c)] and matched reduced [(d)–(f)] cloak, after steady state is reached. The amplitude scale is normalized such that 1 is the maximum plane wave field amplitude without any device present. (a) and (d) At 1.7 GHz, a PEC wall is formed inside the cloaking shell where fields cannot penetrate. The devices behave as a conductive scatterer and a large shadow is observed behind them. (b) and (e) At 2.0 GHz, the devices operate at their nominal frequency. For (e) only, imperfections in the field distributions are inherent in the limitations of the cloak’s approximate design. (c) and (f) At 2.3 GHz, the cloaking material is perceived as a dielectric scatterer from the incident wave. Thus, no significant shadow is formed behind the devices.

Figure 8

(Color online) Time-dependent snapshots of the real part of the magnetic field amplitude distribution $\left(H_z\right)$ when a 1 GHz (FWHM) Gaussian pulse is impinging on dispersive devices. (a)–(c) Ideal cloak. (d)–(f) Ideal concentrator. In (a) and (d), the pulse has reached the first half of the devices. In (b) and (e) the snapshots are taken 1.2 nsec later than (a) and (d), when the pulse is leaving the dispersive regions. The wavefronts near the center of the devices are delayed compared to the wavefronts away from the central regions. In (c) and (f), the snapshots are taken 0.7 nsec later than (b) and (e). A delay appears near the center of the devices, even though the waves have recomposed. Reflections are observed due to the broad bandwidth of the incident pulse.

Figure 9

(Color online) Relative accumulated energy distribution for ideal cloak and concentrator. In the amplitude scale shown, 1 is the accumulated energy distribution of free space propagation. The lines $R_1$ and $R_2$ indicate the inner and outer radii of the devices. The distributions are not perfectly symmetric because of the finite discretization used in the FDTD simulations.