= Total internal reflection
{wiki}
<Total internal reflection> occurs when a <wave> travels from higher to lower <refractive index> above the <critical angle for total internal reflection>, leaving no propagating transmitted <plane electromagnetic wave> normal to the interface.
At an infinite planar interface, <total internal reflection> returns all incident normal <power> in the reflected <plane electromagnetic wave>. Assume homogeneous <dielectrics> with <isotropy>, <linearity>, no <dielectric loss>, <magnetic permeability> $\mu_0$, and positive real <refractive indices> $n_1>n_2$. Above the <critical angle for total internal reflection>, both <TE polarization> and <TM polarization> have unit <optical reflectance> despite the external <evanescent wave>.<Interactive Optics by Sander Konijnenberg, Aurèle J.L. Adam and H. Paul Urbach>[Basic Electromagnetic and Wave Optics, Total Internal Reflection and Evanescent Waves, equations (153)–(158)]{ref}
This makes <total internal reflection> useful for guiding <light> in an <optical fiber>: suitable internal incidence <angles> keep <energy> inside successive boundaries. Its more surprising feature is that perfect <optical reflectance> coexists with an external <electric field>. The <evanescent field in total internal reflection> can interact with a nearby object, as the <frustrated total internal reflection> experiment demonstrates.
For an original idealized example, choose $n_1=1.5$, $n_2=1$ at <vacuum> <wavelength> $\lambda_0=633\,\mathrm{nm}$. At <angle of incidence> $40^\circ$, <Snell law for electromagnetic waves> gives an <angle of refraction> $\arcsin(1.5\sin40^\circ)\approx74.6186^\circ$, so a transmitted <light ray> still emerges. At $\theta_c\approx41.8103^\circ$, its direction becomes tangent to the boundary. At $50^\circ$, the proposed <sine> of a real <angle of refraction> would be $1.5\sin50^\circ\approx1.14907>1$: the transmitted <wavevector> must instead have an imaginary normal component. These are predictions for the stipulated media, not measured properties of a particular block.
\Image[/total-internal-reflection-critical-geometry.png]
{title=Constructed ray geometry just below, at, and above the critical angle for refractive indices 1.5 and 1; arrows indicate directions rather than relative powers}
A real semicircular-block experiment isolates this threshold. The University of Maryland laboratory instructions place a <laser>, motor-driven rotation stage and <photodetector> together; an <aperture> limits the detector voltage. The semicircular block is centered on the rotation axis, with the <photodetector> aligned along its flat face. The instructions identify a strong grazing-beam signal and a weaker signal just before the <critical angle for total internal reflection>. The readout is detector voltage versus stage <angle>. Retracing the incident <laser> beam identifies <normal incidence> for alignment.<Snell’s law and total internal reflection laboratory instructions, University of Maryland Department of Physics>[pp. 1–2, alignment procedure and Total Internal Reflection apparatus and detector instructions]{ref}
The semicircular shape supplies a geometric control: a <light ray> passing radially through the curved face enters at <normal incidence>, avoiding additional entrance <refraction>. The weaker signal near the flat face locates a limiting <angle of refraction>; a vanishing remote-detector signal alone would not establish that the external <electric field> vanishes. Beam width and alignment limit angular resolution; no measured threshold or decay depth is supplied.<Snell’s law and total internal reflection laboratory instructions, University of Maryland Department of Physics>[p. 1, beam-position uncertainty prompt; p. 2, semicircular-block geometry and near-critical signal]{ref}
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