Circularly polarized light (CPL) can induce an asymmetry between the number of left- and right-handed chiral quasiparticles in Dirac and Weyl semimetals. We show that if the photoresponse of the material is dominated by chiral quasiparticles, the total chiral charge induced in the material by CPL can be evaluated in a model-independent way through the chiral anomaly. In the presence of an external magnetic field perpendicular to the incident CPL, this allows to predict the linear density of the induced photocurrent resulting from the chiral magnetic effect. The predicted effect should exist in any kind of Dirac or Weyl materials, with both symmetric and asymmetric band structure. An estimate of the resulting chiral magnetic photocurrent in a typical Dirac semimetal irradiated by an infrared laser of intensity $\simeq 5 \times 10^6, \lambda \simeq 10, \mu$ in an external magnetic field yields a current $J \simeq 50,\simeq 50,\mu$. This current scales linearly with the magnetic field and wavelength, opening up possibilities for applications in photonics, optoelectronics, and THz sensing.

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