Second Harmonic Generation due to the Spatial Structure of a Radiation Beam

We show that spatially structured radiation generates a second harmonic even in homogeneous and isotropic media. The effect originates from the nonlocality of the electric response to the structured electromagnetic field. We develop an analytical theory of such a second harmonic generation in conducting two-dimensional systems. For the general type of structured radiation, we calculate the emerging electric currents at the double frequency and the emitted second harmonic radiation. The theory applied to twisted light reveals that the angular momentum of light doubles in the second harmonic emission. Our results pave the way for light second harmonic generation and structuring beyond the constraints imposed by crystal symmetry.