Joint control of coherent transmission, reflection, and absorption

The authors present here a comprehensive theory for joint coherent control, addressing the challenge of simultaneously manipulating multiple wave properties. Using transmission, reflection, and absorption as illustrative examples, they show that the numerical range provides the underlying mathematical structure governing achievable responses and reveal non-Abelian effects arising from noncommutativity. They further provide a constructive algorithm to realize arbitrary admissible responses. Broadly applicable across wave types and physical observables, this theory underpins applications requiring precise control over multiple wave characteristics.