Symplectic-dilation mixed wavelet transform and its correspondence in quantum optics

The symplectic wavelet transformation [Opt. Lett. 31 (2006) 3432], which is related to quantum optical Fresnel transform, is developed to the symplectic-dilation mixed wavelet transform (SDWT). The SDWT involves both a real-variable dilation-transform and a complex-variable symplectic transform, and possesses well-behaved properties such as Parseval theorem and inversion formula. The entangled-coherent state representation (ESCR) not only underlies the SDWT, but also helps to derive the corresponding quantum transform operator whose counterpart in classical optics is the lens-Fresnel mixed transform [Phys. Lett. A 357 (2006) 163].