Agreement of fractional Fourier optics with the Huygens–Fresnel principle

According to the Huygens–Fresnel principle, the electric field transfer by diffraction from an emitter to a receiver can be split into two diffraction phenomena: from the emitter to an arbitrary intermediate surface and from this surface to the receiver. Expressing field transfers by the usual Fresnel-type integrals complies with this principle. Since a diffraction phenomenon can be mathematically represented by a fractional order Fourier transform, the Huygens–Fresnel principle should accompany the composition of fractional transforms. We show that diffraction representations by fractional Fourier transforms that have been proposed by some authors are not in conformity with the Huygens–Fresnel principle. We also explain how another representation preserves the Huygens–Fresnel principle and should provide a deeper physical insight in the mathematical expression of a diffraction phenomenon, in the framework of a scalar theory.