= Wavefront curvature
{title2=$\kappa=1/R$}
<Wavefront curvature> describes the change in the normal direction across a surface of equal <wave phase>. A plane wavefront has zero curvature; a spherical wavefront has curvature magnitude $1/|R|$ for radius $R$. A general wavefront can have different <principal curvatures> in different transverse directions. The sign convention must be stated when distinguishing converging and diverging waves.<RP Photonics Encyclopedia by Rüdiger Paschotta>[Wavefronts entry, wavefront shapes and changes produced by lenses]{ref}
In a paraxial transverse plane, choose the convention in which a diverging spherical wave has $R>0$. Its quadratic phase variation is approximately
$$
\phi(r)-\phi(0)=\frac{kr^2}{2R},\qquad k=\frac{2\pi}{\lambda}.
$$
The corresponding local ray angle is approximately $r/R$. For illustrative $R=0.50\,\mathrm m$ and $r=2\,\mathrm{mm}$, the angle is $0.004$ radians and the <optical path difference> is $r^2/(2R)=4\,\mathrm{\mu m}$. At wavelength $500\,\mathrm{nm}$, that is eight phase cycles. A modest geometrical departure can therefore produce a substantial optical phase difference.
For an ideal <Gaussian beam> with its waist at $z=0$,
$$
R(z)=z\left[1+\left(\frac{z_R}{z}\right)^2\right],\qquad
\frac1{R(z)}=\frac{z}{z^2+z_R^2}.
$$
The reciprocal form remains finite at the waist: its value is zero, so the wavefront there is locally planar even though the beam radius is minimal. At $z=z_R$, $R=2z_R$. Beam size and curvature are distinct beam parameters.<Propagation of a Gaussian Beam Through a Thin Lens by Francesco D'Angelo>[Radius of Curvature of Gaussian Wavefronts, equation (6); complex parameter, equation (10)]{ref}
A thin lens changes the curvature according to $1/R'=1/R-1/f$ in this convention, while the beam radius at the lens is unchanged in the thin-element approximation.<Propagation of a Gaussian Beam Through a Thin Lens by Francesco D'Angelo>[Equation (16)]{ref} Thus a lens can alter convergence without an immediate jump in the beam width.
Experimentally, curvature can be inferred from a measured phase profile or local wavefront slopes. Phase unwrapping and the chosen reference matter because an intensity image in one plane does not itself provide the quadratic phase. A curved wavefront is not necessarily aberrated: the appropriate ideal reference can itself be spherical or Gaussian-beam curvature.
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