Complex permittivity and attenuation

Use real fields obtained from $E_0e^{ikz-i\omega t}$ and take positive frequency. For a homogeneous local isotropic material with relative permeability equal to one, write the complex index as $\widetilde n=n+i\kappa$. Then its imaginary part gives a factor that decreases with positive depth when $\kappa\ge0$. The sign follows jointly from the time and spatial exponent conventions. Copying an index from a source using the opposite convention without converting its sign would turn loss into growth.

Squaring the index gives two real equations connecting its phase and attenuation components to the real and imaginary permittivity. For an ordinary passive nonmagnetic branch with nonnegative real index, both components can be obtained from the magnitude of permittivity. A lossless negative real permittivity produces a purely imaginary index and an evanescent field. More general magnetic, anisotropic or active media require additional branch analysis; the simple positive-root formulas here are not universal prescriptions for those systems.

The field amplitude decreases exponentially with the imaginary wavenumber. Time-averaged intensity is quadratic in the field amplitude, so it decreases twice as fast in its exponential argument. An absorption coefficient quoted for intensity is therefore twice the inverse amplitude decay length. Stating only a penetration depth without identifying the measured quantity leaves an ambiguity that can produce a factor-of-two error in a thickness specification or an inferred extinction coefficient.

For a single forward wave in a homogeneous medium, the proportionality between intensity and squared electric amplitude is independent of position, so the ratio of intensities is simple even when the wave impedance is complex. This statement does not apply unchanged to a superposition of incident and reflected waves forming a standing pattern. A finite film requires boundary matching and can display interference fringes on top of absorption. The attenuation coefficient by itself describes propagation after the interface has selected the transmitted wave.

At a planar interface some energy can be reflected without being converted to heat. In a passive half-space with a propagating transmitted wave, bulk heating removes the remaining forward energy flux with increasing depth. Energy accounting therefore compares incident, reflected, transmitted and absorbed powers using the actual boundary conditions. A small transmission coefficient is not sufficient evidence for a large bulk imaginary permittivity, especially when the material is highly reflective or the sample contains resonant cavities.

Destructive interference presents another diagnostic trap. Two coherent amplitudes can sum to zero in one observation channel even when the medium is perfectly lossless. Their energy is redistributed among channels rather than necessarily dissipated. The wave lab makes this distinction testable because its controls change phase and relative amplitude while introducing no absorptive material. A genuine extraction of complex permittivity needs phase information or an adequate measurement model, not a single dark spot on a screen.