Beyond Fresnel Wave Surfaces: Theory of Off-Shell Photonic Density of States and Near-Fields in Isotropy-Broken Materials with Loss or Gain

Fresnel wave surfaces, or isofrequency light shells, provide a powerful framework for describing electromagnetic wave propagation in anisotropic media, yet their applicability is restricted to reciprocal, lossless materials and far-field radiation. This paper extends the concept by incorporating near-field effects and non-Hermitian responses arising in media with loss, gain, or non-reciprocity. Using the Om-potential approach to macroscopic electromagnetism, we reinterpret near fields as off-shell electromagnetic modes, in analogy with off-shell states in quantum field theory. Formally, both QFT off-shell states and electromagnetic near-field modes lie away from the dispersion shell; physically, however, wavefunctions of fundamental particles admit no external sources (virtual contributions live only inside propagators), whereas macroscopic electromagnetic near-fields are intrinsically source-generated by charges, currents, and boundaries and are therefore directly measurable—for example via near-field probes and momentum-resolved imaging—making “off-shell” language more natural and operational in our setting. We show that photonic density of states (PDOS) distributions near Fresnel surfaces acquire Lorentzian broadening in non-reciprocal media, directly linking this effect to the Beer–Bouguer–Lambert law of exponential attenuation or amplification. Furthermore, we demonstrate how Abraham and Minkowski momenta, locked to light shells in the far field, naturally shift to characterize source structures in the near-field regime. This unified treatment bridges the gap between sources and radiation, on-shell and off-shell modes, and reciprocal and non-reciprocal responses. The framework provides both fundamental insight into structured light and practical tools for the design of emitters and metamaterial platforms relevant to emerging technologies such as 6G communications, photonic density-of-states engineering, and non-Hermitian photonics.