= Angular spectrum method
{wiki}
= Angular spectrum representation
{synonym}
The \b[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>.<Interactive Optics by Sander Konijnenberg, Aurèle J.L. Adam and H. Paul Urbach>[Scalar Diffraction Optics, Angular Spectrum Method, equations371–375]{ref}
Use <angular frequency> $\omega$ and time dependence $e^{-i\omega t}$, and define $\widetilde U(k_x,k_y;0)=\int\int U(x,y,0)e^{-i(k_xx+k_yy)}\,dx\,dy$. Then
$$
U(x,y,z)=\frac1{(2\pi)^2}\int\int\widetilde U(k_x,k_y;0)e^{i(k_xx+k_yy+k_zz)}\,dk_x\,dk_y,\qquad
k_z=\sqrt{k^2-k_x^2-k_y^2}.
$$
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.<Interactive Optics by Sander Konijnenberg, Aurèle J.L. Adam and H. Paul Urbach>[Equations374–378, branch convention and evanescent fields]{ref}
For an original example, a transverse period $\lambda/2$ gives $k_x=2k$, $k_y=0$ and $\kappa=\sqrt3\,k$. After distance $z=\lambda/4$, its amplitude is multiplied by \b[$e^{-\pi\sqrt3/2}\simeq0.0658$]. 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
$$
U(0,0,z)=\frac{e^{ikz}}{1+iz/z_R},\qquad z_R=kw_0^2/2.
$$
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.<Physical origin of the Gouy phase shift by Simin Feng and Herbert G. Winful>[Equations4–12, transverse confinement and Gaussian phase]{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,$\mathbf k\cdot\widetilde{\mathbf E}=0$. Boundary conditions and source consistency accompany a vector reconstruction.
Back to article page
