Evanescent angular-spectrum component

An evanescent angular-spectrum component has transverse spatial frequency greater than the medium wavenumber. In a homogeneous lossless medium, with $q=\sqrt{k_x^2+k_y^2}>k$, its forward longitudinal wavenumber is imaginary:It oscillates transversely while decaying away from the source plane. The bounded sign is fixed by the outgoing branch of the angular spectrum.^[ref]

For an original transverse period $d=\lambda /2$, take $k_x=2k$, $k_y=0$. Then $\kappa =\sqrt{3}k$ and the $1/e$ amplitude-decay distance is $1/\kappa =\lambda /(2\pi \sqrt{3})\simeq 0.09189\lambda$. At $z=\lambda /4$, the amplitude is about 0.0658 of its source value, while its squared magnitude is about 0.00433. Amplitude and squared-amplitude attenuation must not be interchanged.

This explains a basic limitation of far-field reconstruction: fine transverse structure can be present at a boundary yet be strongly suppressed before reaching a distant detector. Inverting the factor $e^{-\kappa z}$ also amplifies errors. A tiny measured high-frequency component may therefore be an unstable basis for reconstructing fine source detail.

The electromagnetic field is still constrained by Maxwell equations. In particular, $\mathbf{k}\cdot \tilde{\mathbf{E}}=0$ remains valid for complex $\mathbf{k}$. A single ideal evanescent wave in a lossless medium does not carry time-averaged energy flux normally away from its interface in the same way as a propagating wave. Near-field coupling or a superposition involving another boundary can nevertheless transfer energy, so decay is not a universal prohibition on energy exchange.

An illustrative experiment compares the same patterned source with detectors at increasing distances and accounts for the detector's own spatial response. The decay predicted here concerns a resolved spectral component under the stated homogeneous-medium model, not a guarantee that every observed loss of contrast is evanescence. Absorption, finite aperture and instrument resolution can also suppress structure.