Sensitivity-optimal coplanar waveguide design for broadband magnetic resonance spectroscopy: a Beer--Lambert framework

Coplanar waveguide (CPW) transmission spectroscopy is used to probe spin dynamics, ferromagnetic resonance, and complex conductivity across a wide range of materials, yet no systematic framework connects waveguide geometry to measurement sensitivity when sample volume and concentration are fixed. We show that the shared geometric scaling of sample coupling and conductor loss maps CPW design onto the Beer--Lambert optimization problem of optical spectrophotometry, reducing it to a universal one-parameter problem whose solution depends only on sample geometry and the dominant noise source -- not on sample properties, operating frequency, or system losses. The framework predicts a near-universal $1\,\mathrm{Np}$ optimum across the full range of sample thicknesses and design geometries. Benchmarking against seven published broadband FMR instruments reveals two fabrication-delimited classes: PCB-milled designs are bounded by a ceiling imposed by their minimum slot width, while photolithographic designs approach the additive-noise optimum. For large-area PCB samples a meander geometry offers a direct path to near-optimal sensitivity without interferometric compensation; for sub-millimeter samples, a single lithographic straight pass suffices.