Fractional Hankel transform studied by charge-amplitude state representations and complex fractional Fourier transformation

New complex integration transformation and its compatibility with complex Weyl-Wigner transformation and entangled state representation

Enlightened by the special transformation in our preceding paper [J. Mod. Opt. 56, 1227 (2009)], we propose a new complex integration transformation corresponding to two mutually conjugate two-mode entangled states <eta| and <xi| that is compatible with eta-xi phase space quantum mechanics and can be used to obtain the complex fractional Fourier transformation kernel from the chirplet function. This transformation obeys the Parseval theorem and is invertible.

Symplectic wavelet transformation

Usually a wavelet transform is based on dilated-translated wavelets. We propose a symplectic-transformed-translated wavelet family psi(*)(r,s)(z-kappa) (r,s are the symplectic transform parameters, |s|(2)-|r|(2)=1, kappa is a translation parameter) generated from the mother wavelet psi and the corresponding wavelet transformation W(psi)f(r,s;kappa)=integral(infinity)(-infinity)(d(2)z/pi)f(z)psi(*)(r,s)(z-kappa). This new transform possesses well-behaved properties and is related to the optical Fresnel transform in quantum mechanical version.

Collins diffraction formula studied in quantum optics

We find that the Collins diffraction formula in cylindrical coordinates is just the transformation matrix element of a three-parameter two-mode squeezing operator in the deduced entangled state representation. This is a new tie connecting the unitary transform in quantum optics to the generalized Hankel transform in Fourier optics. The group multiplication rule of the squeezing operators maps to the Collins formula related to two successive Hankel transforms.