  Implementation of a parity-time (PT) symmetric microwave photonic system in
the optical wavelength space with spatial singularity is proposed. In the
proposed PT-symmetric microwave photonic system, the gain and loss modes are
confined in a single spatial resonator, which is different from a conventional
PT-symmetric system in which the two modes are localized in two physically
separated resonators to form one-dimensional spatial potential symmetry as
required by the simplest one-dimensional parity transformation. We show that
PT-symmetry can be implemented between subspaces in non-spatial parameter
spaces, in which the gain and loss modes can perfectly overlay spatially but
are distinguishable in the designated parameter space. The resultant spatial
singularity enables the possibility in implementing PT-symmetric systems with
increased structural simplicity, integration density and long-term stability.
To prove the concept, a PT-symmetric optoelectronic oscillator (OEO) in the
optical wavelength space is implemented. The OEO has a single-loop
architecture, with the gain and loss microwave modes carried by two optical
wavelengths to form two mutually coupled wavelength-space resonators (WSRs).
PT-symmetry is achieved by controlling the wavelength spacing and the power
contrast. The operation of PT symmetry in the OEO is verified by the generation
of a 10-GHz microwave signal with a low phase noise of 129.3 dBc/Hz at 10-kHz
offset frequency and a high sidemode suppression ratio (SMSR) of 66.22 dB.
Compared with a conventional spatial PT-symmetric system, one in the wavelength
space features a much simpler configuration, better stability and greater
resilience to environmental interferences.
