We investigate the frequency conversion in a waveguide by dynamically modulating the real and imaginary parts of permittivity, namely, the complex modulation. A single frequency of incident light could convert into many discrete frequencies with equivalent intervals, which is analogous to discrete diffraction in spatial waveguide arrays. We develop a coupled mode theory to analyze the frequency conversion process and show the complex modulation results in a complex-valued coupling coefficient between adjacent frequency channels. The coupling strength can be flexibly tuned by controlling the modulation amplitudes. Moreover, the complex coupling becomes asymmetric with increasing the modulation phase difference between the real and imaginary parts of permittivity. The capability to control the complex coupling leads to new features in discrete diffraction in the frequency dimension, including the unidirectional transport and equalization of light intensities at different channels. The study may find interesting applications in frequency comb generators and frequency controllers.
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