= Outgoing branch of the angular spectrum
The \b[<outgoing branch of the angular spectrum>] selects the longitudinal <wavenumber> that represents forward radiation and bounded evanescent decay in a specified source-free half-space. For real $k>0$, time dependence $e^{-i\omega t}$ and propagation into $z>0$,
$$
k_z=\begin{cases}\sqrt{k^2-q^2},&q<k,\\ i\sqrt{q^2-k^2},&q>k,\end{cases}\qquad q^2=k_x^2+k_y^2.
$$
The first root is positive real and the second positive imaginary. They accompany the factor $e^{ik_zz}$.<Interactive Optics by Sander Konijnenberg, Aurèle J.L. Adam and H. Paul Urbach>[Scalar Diffraction Optics, Angular Spectrum Method, equations 374–378 and square-root convention]{ref}
The scalar <Helmholtz equation> permits both signs. For $q<k$, $e^{ik_zz}$ and $e^{-ik_zz}$ describe opposite longitudinal phase propagation under the stated time convention. For $q>k$, the corresponding factors are decaying and growing exponentials. A source-free upper half-space with no incident field from infinity uses the forward, bounded choice.
For an original transverse spatial frequency $q=1.25k$, the decaying root is $k_z=0.75ik$. At $z=\lambda$, amplitude changes by $e^{-1.5\pi}\simeq0.00898$. Choosing the other sign would instead multiply it by approximately 111.3. A sign error can therefore become an enormous near-field propagation error without changing the input data.
This choice is a statement about the physical region and boundary conditions, not an instruction to delete reflected waves from every problem. An interface or cavity can require both propagation directions in a finite region. Likewise, propagation towards negative $z$ must use a convention consistent with that region. At $q=k$, the root is zero and the component is grazing; formulas that divide by $k_z$ need separate attention.
Backward reconstruction of a decayed <evanescent angular-spectrum component> multiplies measurement error by an increasing exponential. That is an inverse problem's conditioning issue, rather than evidence that the forward outgoing branch should grow. Stating the time convention, physical half-space and radiation condition prevents the square-root symbol from hiding these choices.
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