Transverse modulation in electrovac Brinkmann pp-waves: Maxwell consistency and curvature universality

Electrovac pp–waves in Brinkmann form provide exact Einstein–Maxwell solutions for co–propagating null radiation. Motivated by lensing or scattering, one often ``modulates’’ a plane electromagnetic wave by a weak transverse envelope $1+\gamma f(x,y)$. We show that, within the aligned null pp–wave ansatz ($A_v=0$, no $v$–dependence, $F_{xy}=0$) and enforcing the source–free Maxwell equations to $\mathcal{O}(\gamma )$, a generic profile $f(x,y)$ is incompatible with Maxwell: the transverse field $F_{ui}$ must be both divergence–free and curl–free on the transverse plane, hence $F_{ui}={\mathrm{\partial }}_i\mathrm{\Phi }$ with ${\mathrm{\Delta }}_{\perp }\mathrm{\Phi }=0$. We give a minimal, polarization–agnostic gauge completion of the modulated potential and prove a cancellation theorem: under standard decay/regularity (or zero–mode) conditions that exclude additional harmonic transverse modes, all $\mathcal{O}(\gamma )$ dependence on $f$ drops out of $F_{ui}$ and therefore out of the electrovac source $T_{uu}$. Consequently, the electromagnetic contribution to the Brinkmann profile is universal at $\mathcal{O}(\gamma )$: the familiar cycle–averaged isotropic $r^2$ term plus an isotropic oscillatory correction at frequency $2\omega$, present only for non-circular polarisation. We isolate the residual Maxwell–admissible freedom as harmonic (holomorphic) transverse data and, by Kerr–Schild linearity, superpose an arbitrary co–propagating vacuum gravitational pp–wave, relating TT–gauge strain to Brinkmann amplitudes. Modelling genuinely localised beams, therefore, requires currents, non-null components, or more general Kundt/gyraton geometries.