Plasma lens with frequency-dependent dispersion measure effects on fast radio bursts

Radio signals propagating through inhomogeneous plasma media deviate from their original paths, producing frequency-dependent magnification effects. In this paper, after reviewing the classical plasma-lensing theory, we have found a fundamental contradiction: the classical model assumes that the distribution of lensing plasma medium is related to the frequency-independent image position; however, our analysis demonstrates that both the image position ($\theta(\nu)$) and dispersion measure (DM$(\nu)$) are inherently frequency-dependent when signals traverse a structured plasma medium. We have been able to resolve this paradox by developing a framework that explicitly incorporates frequency-dependent dispersion measures (DMs) following power-law relationships ($\rm DM\propto \nu^\gamma$). Our analysis shows that the signal magnification decreases systematically with decreasing frequency, offering a plausible explanation for the frequency-dependent peak flux densities observed in fast radio bursts (FRBs), particularly in the case of the repeating FRB 180814.J0422+73. Our results suggest these FRBs could originate from the magnetized compact star magnetospheres. By considering these plasma-lensing effects on the sub-pulses of an FRB across different frequencies, we have the ability to more accurately investigate the intrinsic properties of FRBs via precise measurements of radio signals.