of direction), two further conclusions follow: first, the two velocities, and hence their ratio, are independent of their directions; and second, the wave-normal directions coincide with the ray directions, so that θ1 and θ2 coincide with the angles of incidence and refraction as defined above. Obviously the angle of refraction cannot exceed 90°. In the limiting case, we put and in Eq.(), and solve for the critical angle: In deriving this result, we retain the assumption of isotropic media in order to identify θ1 and θ2 with the angles of incidence and refraction. For electromagnetic waves, and especially for light, it is customary to express the above results in terms of refractive indices. The refractive index of a medium with normal velocity is defined as where c is the speed of light in vacuum. Hence Similarly, Making these substitutions in Eqs.()and(), we obtain and Eq.() is the law of refraction for general media, in terms of refractive indices, provided that θ1 and θ2 are taken as the dihedral angles; but if the media are isotropic, then and become independent of direction while θ1 and θ2 may be taken as the angles of incidence and refraction for the rays, and Eq.() follows. So, for isotropic media, Eqs.()and() together describe the behavior in Fig.5. According to Eq.(), for incidence from water () to air (), we have , whereas for incidence from common glass or acrylic () to air (), we have . The arcsin function yielding θc is defined only if Hence, for isotropic media, total internal reflection cannot occur if the second medium has a higher refractive index (lower normal velocity) than the first. For example, there cannot be TIR for incidence from air to water; rather, the critical angle for incidence from water to air is the angle of refraction at grazing incidence from air to water (Fig.6). The medium with the higher refractive index is commonly described as optically denser, and the one with the lower refractive index as optically rarer. Hence it is said that total internal reflection is possible for "dense-to-rare" incidence, but not for "rare-to-dense" incidence. Everyday examples When standing beside an aquarium with one's eyes below the water level, one is likely to see fish or submerged objects reflected in the water-air surface (Fig.1). The brightness of the reflected image – just as bright as the "direct" view – can be startling. A similar