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.
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