EML Operator Representation of Photonic Transfer Functions

The EML operator eml(x,y) = exp(x)−ln(y), recently introduced by Odrzywolek, generates every elementary function from a single binary primitive and the terminal constant 1. We observe that the physical primitives underlying photonic transfer functions—gain, attenuation, phase accumulation, and interface reflection—are themselves compositions of exp and ln, making them structurally natural candidates for EML representation. We derive explicit EML trees for Beer–Lambert attenuation, the linear electro-optic (Pockels) phase shift, and Mach–Zehnder modulator transmission, and give Fresnel coefficients in closed EML form. Our main contribution is the observation that for a planar N-layer stack with generic parameters, the minimal EML-tree node count scales as Θ(N) and the tree depth admits an O(N) upper bound, with each physical layer contributing a bounded increment of symbolic work. This suggests EML-tree size as a candidate complexity measure for optical systems, analogous to circuit depth in digital logic, and motivates EML grammars as structural priors for symbolic regression from FDTD or RCWA data. Coherent modulators, saturable amplifiers, and hybrid optical neural networks fit the same cascaded product structure and inherit the same bound.