%A Hong-Yi Fan, Jun-Hua Chen %T On the core of the fractional Fourier transform and its role in composing complex fractional Fourier transformations and Fresnel transformations %0 Journal Article %D 2015 %J Front. Phys. %J Frontiers of Physics %@ 2095-0462 %R 10.1007/s11467-014-0445-x %P 100301-${article.jieShuYe} %V 10 %N 1 %U {https://journal.hep.com.cn/fop/EN/10.1007/s11467-014-0445-x %8 2015-02-10 %X

By a quantum mechanical analysis of the additive rule Fα[Fβ[f]] = Fα+β[f], which the fractional Fourier transformation (FrFT) Fα[f] should satisfy, we reveal that the position-momentum mutualtransformation operator is the core element for constructing the integration kernel of FrFT. Based on this observation and the two mutually conjugate entangled-state representations, we then derive a core operator for enabling a complex fractional Fourier transformation (CFrFT), which also obeys the additive rule. In a similar manner, we also reveal the fractional transformation property for a type of Fresnel operator.