Angular spectrum method

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 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.

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 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 $z>0$,The first root is positive real and the second positive imaginary. They accompany the factor $e^{i{k}_{z}z}$.^[ref]

The scalar Helmholtz equation permits both signs. For $q<k$, $e^{i{k}_{z}z}$ and $e^{-i{k}_{z}z}$ 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 $e^{-1.5\pi }\simeq 0.00898$. 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.

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.

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.