The Fresnel–fractional Hadamard combinatorial transformation for symmetric beamsplitter

Based on the newly developed coherent-entangled state representation we find so-called Fresnel–fractional Hadamard combinatorial transformation, which is new and unitary. It is remarkable the new operator is concisely expressed in the coherent-entangled state representation as a projective operator in integration form. The new unitary operator plays the role of both Fresnel transformation for (a1−a2)/2 and Fractional Hadamard transformation for (a1+a2)/2, respectively. Physically, (a1−a2)/2 and (a1+a2)/2 could be a symmetric beamsplitter's two output fields.