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For a rigid screen $x<0$, use time dependence $e^{i\omega t}$ and inverse Fourier transform $e^{-i\kappa x}$. The scattered field is odd in $y$. Its upper-face trace is supported on $x<0$, while its normal derivative on $x>0$ is unknown. With the outgoing factor $\gamma =\sqrt{{\kappa }^2-k_0^2}$, these traces satisfy $\gamma {\hat{\varphi }}^{-}+G^{+}=b/(\kappa +a)$, where $a=k_0\cos {\theta }_0$ and $b=k_0\sin {\theta }_0$. Factor $\gamma =\sqrt{\kappa -k_0}\sqrt{\kappa +k_0}$ into upper/lower analytic factors. Pole subtraction in a Wiener-Hopf equation and the finite-energy edge condition give the displayed trace. The forcing-pole residue yields geometrical optics, while stationary phase yields the outgoing edge waveThe angular formula excludes the shadow-boundary transition regions; the exact integral supplies their uniform continuation.

The Wiener-Khinchin theorem relates the autocorrelation of a weakly stationary process to its power spectral density by a Fourier transform. The displayed convention is appropriate when the optical correlation is $\Gamma (\tau )=⟨f^{\ast }(t)f(t+\tau )⟩$ and the analytic field contains $e^{-i\omega t}$. A Lorentzian line shape of half-width $\kappa$ gives exponential correlation decay $e^{-\kappa |\tau |}$, including the carrier factor $e^{-i\omega \tau }$ when the full field is correlated.

The Wigner distribution of two fields is the Fourier transform of a field's two-point product in its separation variable:Its inverse reconstructs $u(\mathbf{z})\overline{v({\mathbf{z}}^{\prime })}$. For vector fields, specify whether the product is an outer product or a scalar contraction; the latter can encode an average over receiver channels.

A window function is a function that vanishes outside a chosen finite interval; multiplying an input by it selects and weights that part of the history before a Fourier transform. The rectangular window $w_T$ is the indicator function of $[0,T]$. Applied to the weighted causal extension by zero of an input, it produces the finite-history integral ${\int }_0^T{e}^{-st}f(t)dt$ used to approximate a Laplace transform.

The spatially constant Fourier mode. On a periodic domain its coefficient represents the spatial mean, up to the chosen Fourier transform normalization. Derivatives annihilate this mode, so homogeneous periodic diffusion preserves it.