  This article demonstrates the realization of an extraordinary beam splitter
based on nonreciprocal and synchronized photonic transitions in obliquely
illuminated space-time-modulated (STM) slabs which impart the coherent temporal
frequency and spatial frequency shifts. As a consequence of such unusual
photonic transitions, a one-way beam splitting and amplification is exhibited
by the STM slab. Beam splitting is a vital operation for various optical and
photonic systems, ranging from quantum computation to fluorescence spectroscopy
and microscopy. Despite the beam splitting is conceptually a simple operation,
the performance characteristics of beam splitters significantly influence the
repeatability and accuracy of the entire optical system. As of today, there has
been no approach exhibiting a nonreciprocal beam splitting accompanied with
transmission gain and an arbitrary splitting angle. Here, we show that oblique
illumination of a periodic and semi-coherent dynamically-modulated slab results
in coherent photonic transitions between the incident light beam and its
counterpart space-time harmonic (STH). Such photonic transitions introduce a
unidirectional synchronization and momentum exchange between two STHs with same
temporal frequencies, but opposite spatial frequencies. Such a beam splitting
technique offers high isolation, transmission gain and zero beam tilting, and
is expected to drastically decrease the resource and isolation requirements in
optical and photonic systems. In addition to the analytical solution, we
provide a closed-form solution for the electromagnetic fields in STM
structures, and accordingly, investigate the properties of the wave isolation
and amplification in subluminal, superluminal and luminal ST modulations.
