Amplification in an Ideal TL with Temporally Modulated Zo: 1-D Photonic Time Crystal

This paper investigates and simulates signal amplification in an ideal transmission line (TL) with temporal modulated characteristic impedance Zo. First, a lossless L-C TL lumped model is considered. The shunt capacitor with infinite quality factor (Q) is modulated with time in a sinusoidal manner. The eigenvalue problem is formulated, solved, and discussed. The dispersion diagram (DD) is plotted, and the momentum band gap (MBG) is observed at a modulation frequency double the signal frequency. Signal amplification is confirmed using Harmonics Balance (HB) and Transient simulation (TS), and the obtained results are consistent with the DD results. Second, an ideal TL loaded with shunt temporal modulated capacitors (TMCs) with infinite Q is studied. Varying the TL length allows controlling the frequency band gap (FBG) location relative to the MBG in the DD. As the FBG approaches the MBG, wider MBGs can be obtained. As a result, higher values of amplification are expected. The wider obtained MBG is explained by studying the Bloch impedance (ZBloch) of the unit cell. While modulation of the TMC, the TL length affects the ZBloch varying range. The DD and ZBloch results are confirmed through transient and HB simulations. Finally, a series resistance (Rc) is added to the TMC (finite Q). The effect of varying Rc on the DD and signal amplification is studied. As expected, increasing the Rc value increases the charging and discharging time of the TMC, decreasing the system's responsiveness to the modulation and increasing the loss. Consequently, lower values of amplification are obtained.