= Wavefront
{wiki}
A <wavefront> connects points with the same <wave phase> at one specified time. It is a surface, rather than the trajectory of a particular piece of <light>. For a scalar <monochromatic wave> written as $u(\mathbf r,t)=A(\mathbf r)\cos\Phi(\mathbf r,t)$, a <wavefront> is a connected <level set> of the <phase function> $\Phi$; different surfaces can share the same <wave phase> modulo $2\pi$. A <plane wave> has planar <wavefronts>. An outward wave from an ideal point source in a uniform medium has <spheres> as <wavefronts>, while an <optical lens> can turn nearly planar <wavefronts> into converging ones. The <wavefront> describes <wave phase>, not <wave amplitude>: drawing it does not say how bright the beam is. <RP Photonics Encyclopedia by Rüdiger Paschotta/Wavefronts>{ref}
The relation to <light rays> becomes precise in a stationary medium with <isotropy>, negligible loss and a real positive scalar <refractive index> $n(\mathbf r,\omega)$. In the <geometrical optics> limit, spatial changes in the <refractive index> and <wave amplitude> are slow on the local <wavelength> scale. For a single smooth propagation branch, write
$$
\Phi(\mathbf r,t)=k_0S(\mathbf r,\omega)-\omega t+\phi_0,\qquad k_0=\frac{\omega}{c},\qquad |\nabla S|^2=n^2.
$$
Here $\omega>0$ is the <angular frequency>, $c$ is the <vacuum> <speed of light>, $S$ has units of length, and the last relation is the optical <eikonal equation>. The local <wavevector> is $\mathbf k=\nabla\Phi=k_0\nabla S$. Its direction is normal to a smooth <wavefront>, since a <gradient> is normal to its <level sets>. A forward <light ray> has <unit tangent vector> $d\mathbf r/ds=\nabla S/n$, so it follows this normal under the stated assumptions. Along that <light ray>, $dS/ds=\nabla S\cdot(d\mathbf r/ds)=n$: the change in $S$ is the <optical path length>. This is the short geometric argument linking <wavefronts>, <light rays> and <optical path length>. <MIT 2.71 and 2.710 Optics, Spring 2014 by Nicholas X. Fang/Optics, Lecture 10: Eikonal Equations, Superposition of EM Waves>{ref}
\Image[/wavefront-refraction-phase-and-rays.png]
{title=<Wavefronts> and normal <light rays> at an interface: an original <plane wave> model of <refraction>}
{id=wavefront-refraction-phase-and-rays}
The diagram shows incident and transmitted branches separately, with the reflected branch omitted. Its stipulated <refractive indices> are $n_1=1$ and $n_2=3/2$, and its incident <angle> is $30^\circ$ to the interface normal. At a stationary interface, the <frequency> and tangential <wavevector> agree on the two sides: <Snell's law> gives $\sin\theta_2=1/3$, hence $\theta_2\approx19.4712^\circ$. The <wavefronts> turn while their normal spacing decreases from $\lambda_0$ to $2\lambda_0/3$, where $\lambda_0$ is the <vacuum> <wavelength>. More explicitly, with $x$ normal to the interface and $y$ tangential, take $S_1=\sqrt3x/2+y/2$ for $x<0$ and $S_2=\sqrt2x+y/2$ for $x>0$. These functions agree at $x=0$; the drawn <wavefronts> are $S_j=m\lambda_0$ at $t=0$ with $\phi_0=0$. This is an exact piecewise <plane wave> phase construction for the stipulated transparent media, not measured material data. <Electromagnetism lecture notes by Richard Fitzpatrick/Laws of Geometric Optics>{ref}
The normal-to-<wavefront> rule requires its assumptions. In a medium with <anisotropy>, the <wavevector> remains normal to a smooth <wavefront>, but the <group velocity> and <energy flux> need not follow it. <RP Photonics Encyclopedia by Rüdiger Paschotta/Wavefronts>{ref} Near a <wave caustic>, where multiple <light rays> meet, one smooth <phase function> may no longer describe the field; <diffraction> and the full <wave amplitude> matter. A <wavefront> is also not generally a surface of equal pulse arrival time. In a medium with <optical dispersion>, a monochromatic <wave phase> moves with the <phase velocity>, while a sufficiently narrow <wave packet> follows the <group velocity>. Even identical <optical path lengths> at one <frequency> can give different <group delays>. The usual equal-travel-time picture needs a common launch reference and a nondispersive ray model; an arbitrary broadband disturbance has no single monochromatic <wavefront>.<The Feynman Lectures on Physics, Volume I by Richard Feynman and others>{ref}
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