The generalized Fresnel–Hadamard complementary transformation for asymmetric beamsplitter

Based on the newly developed parameterized coherent-entangled state representation we propose so-called the generalized Fresnel–Hadamard complementary transformation for asymmetric beamsplitter, which is unitary. The new unitary operator plays the role of both Fresnel transformation for a1 sin θ − a2 cos θ and Hadamard transformation for a1 cos θ + a2 sin θ, respectively. Physically, a1 sin θ − a2 cos θ and a1 cos θ + a2 sin θ could be a asymmetric beamsplitter’s two output fields. We show that the two transformations are concisely expressed in the parameterized coherent-entangled state representation as a projective operator in integration form.