We propose a feasible invisibility approach to suppress the scattering of waves from or to given directions and for particular frequencies, i.e., invisibility on demand. We derive a generalized Hilbert transform for a specific invisibility arrangement relating the two quadratures of the complex permittivity of an object. The scheme allows either designing objects to be invisible, or alternatively modifying the complex susceptibility of a given object to render invisibility. The theoretical proposal is further confirmed by finite-difference time-domain numerical calculations. Moreover, following an iterative chain of generalized Hilbert transforms, we propose the invisibility on demand with additional constraints, i.e., restricting the required modification of the complex refractive index within practical limits by avoiding gain areas. The proposed concept not only opens a venue of real or complex valued index landscapes, but also bridges the gap between Hermitian and non-Hermitian optical systems by unifying these systems under one single generalized design theory.

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