A Note on Fractional Curl Operator

In this letter, we demonstrate that the fractional curl operator, widely used in electromagnetics since 1998, is essentially a rotation operation of components of the complex Riemann-Silberstein vector representing the electromagnetic field. It occurs that after the wave decomposition into circular polarisations, the standard duality rotation with the angle depending on the fractional order is applied to the left-handed basis vector while the right-handed basis vector stems from the complex conjugation of the left-handed counterpart. Therefore, the fractional curl operator describes another representation of rotations of the electromagnetic field decomposed into circular polarisations. Finally, we demonstrate that this operator can describe a single-qubit phase-shift gate in quantum computing.