 Propagating angular-spectrum component

A propagating angular-spectrum component is a transverse Fourier transform component whose longitudinal wavenumber is real in a homogeneous lossless medium. With $q^2=k_x^2+k_y^2<k^2$ and the forward outgoing branch of the angular spectrum,Its propagation factor $e^{i{k}_{z}z}$ has unit magnitude and a distance-dependent wave phase.^[ref]

For an original component with $k_x=0.6k$ and $k_y=0$, $k_z=0.8k$ and $\theta \simeq 36.87{}^{\circ }$. It repeats its phase along $z$ over distance $2\pi /k_z=1.25\lambda$, even though its wavelength measured along its propagation direction remains $\lambda$. The axial phase period is not an energy-transit time.

Preserving each component's magnitude does not preserve the reconstructed spatial intensity. Different values of $k_z$ accumulate different phases; their coherent superposition changes the interference pattern and can spread a beam. This is the angular-spectrum description of diffraction.

Energy crossing a reference plane also depends on direction. For a vacuum plane electromagnetic wave with electric-field magnitude $E_0$, the time-averaged Poynting vector has magnitude $|E_0{|}^2/(2Z_0)$ along the propagation direction. Its normal component through a plane of constant $z$ is that magnitude times $\cos \theta$, where $Z_0$ is the vacuum impedance. Equal electric-field magnitudes at angles zero and $60{}^{\circ }$ therefore give normal fluxes in the ratio $1:1/2$.

The boundary case $q=k$ is grazing, with $k_z=0$ and no forward normal phase advance. For $q>k$, the root becomes imaginary and the component is instead an evanescent angular-spectrum component. These classifications apply to the stated real-wavenumber, lossless model; material absorption requires a consistent complex-wavevector treatment.