Which Waveguide Network Realizes a Prescribed Transmission Profile? An Exact Forward Construction

We introduce an analytically invertible framework for wavefront construction based on the scattering properties of periodic waveguide networks governed by a gauge-shifted Helmholtz operator. By determining the exact transmission coefficients of the network, we express the lattice reactance as a Fourier expansion whose coefficients are analytically mapped onto the underlying graph architecture, allowing the required bond connections, refractive indices, lengths, and gauge phases to be determined directly from a prescribed target transmission coefficient. In contrast to conventional inverse-design approaches, the present formulation provides a closed-form route from desired wave transmission profiles to physically realisable structures. The framework extends naturally from one-dimensional angular filtering to two-dimensional image synthesis, where arbitrary transmitted intensity patterns are reconstructed through exact spectral control of the network scattering response.