  The Josephson effects associated with quantum tunneling of Cooper pairs
manifest as nonlinear relations between the superconductivity phase difference
and the bias current and voltage. Many novel phenomena appear, such as Shapiro
steps in dc cuurent-voltage (IV) characteristics of a Josephson junction under
microwave shining, which can be used as a voltage standard. Inversely, the
Josephson effects provide a unique way to generate high-frequency
electromagnetic (EM) radiation by dc bias voltage. The discovery of cuprate
high-Tc superconductors accelerated the effort to develop novel source of EM
waves based on a stack of atomically dense-packed intrinsic Josephson junctions
(IJJs), since the large superconductivity gap covers the whole terahertz
frequency band. Very recently, strong and coherent terahertz radiations have
been successfully generated from a mesa structure of
$\rm{Bi_2Sr_2CaCu_2O_{8+\delta}}$ single crystal which works both as the source
of energy gain and as the cavity for resonance. It is then found theoretically
that, due to huge inductive coupling of IJJs produced by the nanometer junction
separation and the large London penetration depth of order of $\rm{\mu m}$ of
the material, a novel dynamic state is stabilized in the coupled sine-Gordon
system, in which $\pm \pi$ kinks in phase differences are developed responding
to the standing wave of Josephson plasma and are stacked alternatively in the
c-axis. This novel solution of the inductively coupled sine-Gordon equations
captures the important features of experimental observations. The theory
predicts an optimal radiation power larger than the one available to date by
orders of magnitude, and thus suggests the technological relevance of the
phenomena.
