Active Microwave Cloaking Using Parity-Time-Symmetric Satellites

In this paper, we present and study the dynamics of an active microwave cloaking technique that alleviates the trade-offs between loss, bandwidth, and size inherent in passive approaches. The presented cloak consists of a finite number of satellite antennas loaded with parity-time-symmetric lumped admittances, which, without external control, can track in real time the amplitude, frequency, and phase variations of the impinging signals and generate the required current distributions at the antennas to totally suppress the scattered waves, including reflections and shadows. We analytically derive the optimal density and frequency dispersion of the antenna satellites surrounding the object in order to achieve substantial reduction in the scattering crosssection over a wide bandwidth. We also investigate the conditions necessary to maintain the cloak stability and study its transient response for excitation with pulsed signals.

Figure 1

TM^z plane-wave incidence on a circular PEC cylinder surrounded by N satellite antennas with wire admittance $Y_k$. (a) Geometry of the problem. (b) Implementation of the satellite antennas as thin wires periodically loaded with lumped admittance $Y_{L,k}$.

Figure 2

Optimal current distribution at $f_0$ to achieve $\overline{\mathrm{RCS}}=-30\mathrm{dB}$ for $a={\lambda }_0/2$, $r_c=a+0.05{\lambda }_0$, and N = 10.

Figure 3

Normalized scattering coefficient $\overline{\mathrm{RCS}}$ vs the number of satellites N for different sizes of the PEC cylinder.

Figure 4

Uncloaked scattering crosssection ${\overline{\sigma }}_b$ and the required number of satellites N to achieve $\overline{\mathrm{RCS}}=-30\mathrm{dB}$ vs the size of the PEC cylinder.

Figure 5

Magnitude of the scattering harmonics with and without cloaking for $a={\lambda }_0/2$ and different numbers of the satellites: (a) $N=2$, (b) $N=5$, (c) $N=8$, and (d) $N=10$.

Figure 6

Optimal values of the modified wire admittance $Y_k^{\mathrm{\prime }}$ at $f_0$ for $a={\lambda }_0/2$, $r_c=a+0.05{\lambda }_0$ and N = 10.

Figure 7

Total electric field distribution (normalized to $E_0$) for the same parameters as in Fig. 6: (a) Without cloaking satellites. (b) With cloaking satellites.

Figure 8

Normalized scattering coefficient $\overline{\mathrm{RCS}}$ versus the angle of incidence ${\varphi }_{\mathrm{inc}}$ at $f_0$ for the same parameters as in Fig. 6.

Figure 9

Cloaked scattering crosssection ${\overline{\sigma }}_c$ versus the azimuthal angle $\varphi$ for different angles of incidence: (a) ${\varphi }_{\mathrm{inc}}=0^{\circ }$, (b) ${\varphi }_{\mathrm{inc}}={60}^{\circ }$, (c) ${\varphi }_{\mathrm{inc}}={120}^{\circ }$, and (d) ${\varphi }_{\mathrm{inc}}={180}^{\circ }$. The parameters of the structure are the same as in Fig. 6.

Figure 10

(a) Closed-loop feedback MIMO model for the proposed PT cloak. (b) Open-loop MIMO model for the active cloak in Ref. [24].

Figure 11

Optimal dispersion (solid) for the gain element at $\varphi =0^{\circ }$ to maintain a flat 30-dB reduction compared to the actual dispersion (dashed) achieved with the circuit implementation in Fig. 12.

Figure 12

Circuit implementation of the kth satellite. (a) Passive side. (b) Gain side. A simplified schematic (dc biasing is not shown) for the NIC circuit is also shown using two cross-coupled CMOS transistors [40, 41].

Figure 13

Stability function $H(\omega )$ as defined in Eq. (33) where the bright spots correspond to the poles of the system for different values of Q: (a) $Q=0.1$ where unstable poles can be found in the lower-half plane. (b) $Q=0.5$ where all poles lie in the upper-half plane indicating stability.

Figure 14

Scattering coefficient $\overline{\mathrm{RCS}}$ vs frequency for the same parameters as in Fig. 13.

Figure 15

Transient response of the generated current in the gain element at $\varphi ={72}^{\circ }$ due to an incident plane wave carrier, the amplitude of which is digitally modulated by an on-off pulse: (a) Rising edge of the envelope. (b) Falling edge of the envelope.

Figure 16

Snapshots for the total electric field (normalized to $E_0$) at the rising edge of an amplitude-modulated sinusoidal signal impinging on a cloaked PEC cylinder with radius $a={\lambda }_0/2$.

Figure 17

Snapshots for the total electric field (normalized to $E_0$) at the falling edge of an amplitude-modulated sinusoidal signal impinging on a cloaked PEC cylinder with radius $a={\lambda }_0/2$.