page_title: Total internal reflection
section_title: Laplace, Malus, and attenuated total reflectance (ATR)
hierarchical_section_title: Total internal reflection / History / Laplace, Malus, and attenuated total reflectance (ATR)
caption_attribution_description: Français&#160;: Pierre-Simon de Laplace, mathématicien et astronome, en habit de chancelier du Sénat sous l'Empire
context_page_description: Total internal reflection is the optical phenomenon in which the surface of the water in a fish-tank when viewed from below the water level, reflects the underwater scene like a mirror, with no loss of brightness. In general, TIR occurs when waves in one medium reach the boundary with another medium at a sufficiently slanting angle, provided that the second medium is transparent to the waves and allows them to travel faster than in the first medium. TIR occurs not only with electromagnetic waves such as light and microwaves, but also with other types of waves, including sound and water waves. In the case of a narrow train of waves, such as a laser beam, we tend to describe the reflection in terms of "rays" rather than waves. In a medium whose properties are independent of direction, such as air, water, or glass, each "ray" is perpendicular to the associated wavefronts.
  Refraction is generally accompanied by partial reflection. When waves are refracted from a medium of lower propagation speed to a medium of higher propagation speed, the angle of refraction is greater than the angle of incidence.
context_section_description: William Hyde Wollaston, in the first of a pair of papers read to the Royal Society of London in 1802, reported his invention of a refractometer based on the critical angle of incidence from an internal medium of known "refractive power" (refractive index) to an external medium whose index was to be measured. With this device, Wollaston measured the "refractive powers" of numerous materials, some of which were too opaque to permit direct measurement of an angle of refraction. Translations of his papers were published in France in 1803, and apparently came to the attention of Pierre-Simon Laplace.
  According to Laplace's elaboration of Newton's theory of refraction, a corpuscle incident on a plane interface between two homogeneous isotropic media was subject to a force field that was symmetrical about the interface. If both media were transparent, total reflection would occur if the corpuscle were turned back before it exited the field in the second medium. But if the second medium were opaque, reflection would not be total unless the corpuscle were turned back before it left the first medium; this required a larger critical angle than the one given by Snell's law, and consequently impugned the validity of Wollaston's method for opaque media. Laplace combined the two cases into a single formula for the relative refractive index in terms of the critical angle (minimum angle of incidence for TIR). The formula contained a parameter which took one value for a transparent external medium and another value for an opaque external medium. Laplace's theory further predicted a relationship between refractive index and density for a given substance.
  In 1807, Laplace's theory was tested experimentally by his protégé, Étienne-Louis Malus. Taking Laplace's formula for the refractive index as given, and using it to measure the refractive index of bees' wax in the liquid (transparent) state and the solid (opaque) state at various temperatures (hence various densities), Malus verified Laplace's relationship between refractive index and density.
  But Laplace's theory implied that if the angle of incidence exceeded his modified critical angle, the reflection would be total even if the external medium was absorbent. Clearly this was wrong: in Eqs. (12) above, there is no threshold value of the angle θᵢ beyond which κ becomes infinite; so the penetration depth of the evanescent wave (1/κ) is always non-zero, and the external medium, if it is at all lossy, will attenuate the reflection. As to why Malus apparently observed such an angle for opaque wax, we must infer that there was a certain angle beyond which the attenuation of the reflection was so small that ATR was visually indistinguishable from TIR.
caption: Pierre-Simon Laplace (1749–1827).
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