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The angular spectrum representation expresses a monochromatic wave as a superposition of plane waves. In a homogeneous, lossless, source-free half-space with real wavenumber $k=2\pi /\lambda >0$, transverse Fourier transform components propagate independently, providing a useful solution of the Helmholtz equation.^[ref]

Use angular frequency $\omega$ and time dependence $e^{-i\omega t}$, and define $\tilde{U}(k_x,k_y;0)=\int \int U(x,y,0)e^{-i(k_{x}x+k_{y}y)}dxdy$. Then
The outgoing branch of the angular spectrum has nonnegative real and imaginary parts of $k_z$ for propagation toward $z>0$. A propagating angular-spectrum component has real $k_z$; an evanescent angular-spectrum component has $k_z=i\kappa$ and decays as $e^{-\kappa z}$. The source-plane amplitudes and their complex arguments matter, not just their squared magnitudes.^[ref]

For an original example, a transverse period $\lambda /2$ gives $k_x=2k$, $k_y=0$ and $\kappa =\sqrt{3}k$. After distance $z=\lambda /4$, its amplitude is multiplied by $e^{-\pi \sqrt{3}/2}\simeq 0.0658$. Such fine spatial detail does not propagate with unchanged amplitude to the far field.

The paraxial angular-spectrum propagator uses $k_z\simeq k-(k_x^2+k_y^2)/(2k)$. For the source-plane Gaussian beam $U_0=e^{-(x^2+y^2)/w_0^2}$, its transverse spectrum is proportional to $e^{-w_0^2(k_x^2+k_y^2)/4}$. The on-axis Gaussian integral therefore gives
The denominator produces the axial Gouy phase $-\arctan (z/z_R)$. This connects the phase shift with interference of transverse spectral components, rather than an abrupt phase jump at the focus.^[ref]

The scalar propagation formula does not authorize arbitrary independent vector components. A physical electromagnetic field must satisfy the transverse polarization constraint in an angular spectrum; in a uniform source-free medium,$\mathbf{k}\cdot \tilde{\mathbf{E}}=0$. Boundary conditions and source consistency accompany a vector reconstruction.

The paraxial angular-spectrum propagator is the quadratic-phase approximation to forward free propagation of transverse Fourier transform components concentrated near the optical axis. For real medium wavenumber $k>0$, write $q^2=k_x^2+k_y^2$. With time convention $e^{-i\omega t}$, the exact factor is $e^{iz\sqrt{k^2-q^2}}$. Expanding at $q/k\ll 1$ givesafter removing the common carrier $e^{ikz}$.^[ref]

The approximation preserves the magnitude of each retained spectral component and changes its wave phase. Recombining those components produces diffraction; it need not preserve the shape or intensity at an individual spatial point. In coordinate space the envelope satisfies the parabolic wave equation, $2ik{\partial }_{z}U+{\nabla }_{\perp }^{2}U=0$.

Small propagation angle and acceptable accumulated phase error are separate conditions. The first omitted term produces a phase discrepancy of order $z{q}^4/(8k^3)$. For an original component with $q/k=0.1$, the exact difference between approximate and exact phases isIt is about 0.00789 radians at $z=100\lambda$, but about 0.789 radians at $z=10,000\lambda$. The direction is still only about $5.74{}^{\circ }$ off axis. Propagation distance matters when phase accuracy is the observable.

For a source-plane Gaussian beam $U_0=\exp [-r^2/w_0^2]$, the spectral width is of order $1/w_0$. Thus $k{w}_0\gg 1$ concentrates the important components in the paraxial region. That provides a reason for the familiar wide-waist condition rather than treating all Gaussian formulas as exact electromagnetic solutions.

A calculation containing substantial evanescent angular-spectrum components or large-angle components requires the exact branch and, when appropriate, the vector transverse polarization constraint in an angular spectrum. The paraxial transfer factor has unit magnitude even at large $q$, so extending it blindly into the evanescent region replaces physical decay by an artificial phase rotation.

The transverse polarization constraint in an angular spectrum requires each source-free electromagnetic plane wave component to have electric field orthogonal to its wavevector in the algebraic sense:For an isotropic homogeneous medium with constant permittivity, this follows by inserting a Fourier component into $\nabla \cdot \mathbf{D}=0$. It restricts the vector fields that can accompany a scalar angular spectrum method calculation.^[ref]

With time dependence $e^{-i\omega t}$ and $\mathbf{k}=(k_x,k_y,k_z)$, the relation becomesWhen $k_z\ne 0$, the longitudinal field is therefore ${\tilde{E}}_z=-(k_x{\tilde{E}}_x+k_y{\tilde{E}}_y)/k_z$. It is not an arbitrary third scalar image that can be propagated independently of the first two.

For an original propagating angular-spectrum component with $\mathbf{k}=k(0.6,0,0.8)$, choosing ${\tilde{E}}_x=1$ and ${\tilde{E}}_y=0$ forces ${\tilde{E}}_z=-0.75$. Polarization along $y$ instead gives a purely transverse-to-axis field for this propagation plane. Thus electromagnetic transversality relative to $\mathbf{k}$ does not require zero field along the laboratory $z$ axis.

For an evanescent angular-spectrum component with $\mathbf{k}=k(1.25,0,0.75i)$, the same choice ${\tilde{E}}_x=1$ gives ${\tilde{E}}_z=(5/3)i$. The dot product is not a Hermitian inner product: there is no complex conjugation of $\mathbf{k}$ in the Fourier-transformed divergence equation. The imaginary longitudinal amplitude represents a phase relationship, not permission to discard the constraint.

At grazing propagation, $k_z=0$, division by $k_z$ fails and the original equation must be used. Boundary conditions, material response and the corresponding magnetic field remain necessary to reconstruct a physical solution. In the strongly paraxial approximation, longitudinal components may be small, but that is an approximation derived from the allowed spectrum rather than a replacement for the vector condition.