= Geometrical optics
{wiki}
= Geometric optics
{synonym}
Geometrical optics describes propagation by rays and <Snell's law>, neglecting <diffraction> on scales much larger than the <wavelength>.
A <light ray> is a path in an approximation, not a physical thread of zero width. In a transparent, homogeneous, isotropic medium it is straight and perpendicular to a <wavefront>; boundaries redirect it by the <law of reflection> or <Snell's law>. The approximation is useful when the <wavelength> is small compared with relevant apertures, surface curvature scales, and distances over which the <refractive index> changes. For an illustrative <wavelength> of $500\,\mathrm{nm}$ and a $10\,\mathrm{mm}$ aperture, their ratio is $5\times10^{-5}$. That small ratio supports <ray tracing> for image location, but does not eliminate <diffraction> when calculating the fine structure of a focused spot. At a <wave caustic>, <geometrical optics> can fail despite large optical elements. <MIT 2.71 and 2.710 Optics, Spring 2014 by Nicholas X. Fang/Lecture Notes on Geometrical Optics>{ref}.
The mechanism of <refraction> is a change in <phase velocity>. For transparent isotropic media at the chosen <frequency>, the <refractive index> is
$$
n=\frac{c}{v_{\mathrm{phase}}},\qquad n_1\sin\theta_1=n_2\sin\theta_2,
$$
where $c$ is the vacuum <speed of light> and the <angles> are measured from the interface <normal vector>. At a stationary planar interface the <frequency> and tangential <wavevector> match across the boundary. Since the <wavenumber> is $k=n\omega/c$, that matching gives <Snell's law>. The transmitted <wavelength> changes while the <frequency> stays fixed. Here $n$ uses <phase velocity>, which need not equal <group velocity>. https://openstax.org/books/university-physics-volume-3/pages/1-3-refraction[OpenStax: Refraction]{ref}. https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/ce6075363fa304e0a61078f7048e96f9_maxwell_eq_iso.pdf[MIT: Classical electromagnetism and optics, phase and group velocities]{ref}.
Choose model air with $n_1=1$ and model glass with $n_2=1.5$, without claiming these are exact values for every sample or <wavelength>. At $\theta_1=45^\circ$, <Snell's law> gives
$$
\theta_2=\arcsin\!\left(\frac{\sin45^\circ}{1.5}\right)\approx28.1255^\circ.
$$
The transmitted <light ray> bends toward the <normal vector>; its <phase velocity> is $c/1.5$, and its <wavelength> is two thirds of the air value. By <reversibility of an optical ray>, incidence from glass at $28.1255^\circ$ returns the outgoing air <angle> $45^\circ$.
Incidence from glass at $45^\circ$ is a different experiment. It would require $\sin\theta_2=1.5\sin45^\circ\approx1.06066$, so there is no real transmitted <angle>. The <critical angle for total internal reflection> is
$$
\theta_c=\arcsin(1/1.5)\approx41.8103^\circ.
$$
Above this <angle>, an ideal lossless glass-air boundary produces <total internal reflection>. The <physical optics> description still permits an <evanescent wave> in the air, although no propagating transmitted <light ray> crosses the boundary. https://openstax.org/books/university-physics-volume-3/pages/1-4-total-internal-reflection[OpenStax: Total internal reflection]{ref}. https://web.mit.edu/2.710/Fall06/2.710-wk7-b-ho.pdf[MIT optics: Total internal reflection and evanescent waves, slide 40]{ref}.
\Image[/optical-refraction-rays.png]
{title=Refraction into glass and total internal reflection on incidence from glass; all angles are measured from the normal}
The usefulness of <geometrical optics> is that complicated instruments can be assembled from these local rules. Applying <Snell's law> at successive curved surfaces changes a spreading bundle into a converging one. The <Fermat principle> unifies the construction through stationary <optical path length>; stationarity does not require a global minimum. https://www.feynmanlectures.caltech.edu/I_26.html[Feynman Lectures: The principle of least time, section 26-5]{ref}. Fang's MIT notes explicitly connect this principle to <specular reflection>, <refraction>, and <thin lens> imaging, providing a route from individual paths to complete instruments. <MIT 2.71 and 2.710 Optics, Spring 2014 by Nicholas X. Fang/Lecture Notes on Geometrical Optics>{ref}.
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