The Fresnel–Weyl complementary transformation

Based on the newly developed coherent-entangled state representation, we propose the so-called Fresnel-Weyl complementary transformation operator. The new operator plays the roles of both Fresnel transformation (for (a₁-a₂)√2 and the Weyl transformation (for (a₁+a₂)√2. Physically, (a₁-a₂)√2 and (a₁+a₂)√2 could be a symmetric beamsplitter's two output fields for the incoming fields a₁ and a₂. We show that the two transformations are concisely expressed in the coherent-entangled state representation as a projective operator in the integration form.