Eigenmodes of fractional Hankel transform derived by the entangled-state method

Discrete fractional Fourier transform as a fast algorithm for evaluating the diffraction pattern of pulsed radiation

A technique is proposed for computing the field radiated from a rectangular aperture. This technique, based on the discrete fractional Fourier transform, avoids the complexities of computing the diffraction pattern by the direct evaluation of the Fresnel integral. The advocated approach provides a fast and accurate computational tool, especially in the case of evaluating pulsed fields radiated through two-dimensional screens of complex amplitude. A detailed numerical study that demonstrates the efficacy of this approach is carried out.

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.

Talbot effect in a quadratic-index medium studied with two-variable Hermite polynomials and entangled states

We have found the new eigenmodes, two-variable Hermite polynomials, exist in propagating plane waves in quadratic-index media and that a two-dimensional Talbot effect can be demonstrated with these modes.

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

Starting from a complex fractional Fourier transformation [Opt. Lett. 28, 680 (2003)], it is shown that the integral kernel of a fractional Hankel transformation is equivalent to the matrix element of an appropriate operator in the charge-amplitude state representations; i.e., the fractional Hankel transformation is endowed with a definite physical meaning (definite quantum-mechanical representation transform).