Wavefront curvature ($\kappa =1/R$)

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.^[ref]

In a paraxial transverse plane, choose the convention in which a diverging spherical wave has $R>0$. Its quadratic phase variation is approximatelyThe 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 $r^2/(2R)=4\mu \mathrm{m}$. 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 $z=0$,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.^[ref]

A thin lens changes the curvature according to $1/R^{\prime }=1/R-1/f$ in this convention, while the beam radius at the lens is unchanged in the thin-element approximation.^[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.