verified that four total internal reflections at 68°27' would give an accurate circular polarization if two of the reflections had water as the external medium while the other two had air, but not if the reflecting surfaces were all wet or all dry. Fresnel's deduction of the phase shift in TIR is thought to have been the first occasion on which a physical meaning was attached to the argument of a complex number. Although this reasoning was applied without the benefit of knowing that light waves were electromagnetic, it passed the test of experiment, and survived remarkably intact after James Clerk Maxwell changed the presumed nature of the waves. Meanwhile, Fresnel's success inspired James MacCullagh and Augustin-Louis Cauchy, beginning in 1836, to analyze reflection from metals by using the Fresnel equations with a complex refractive index. The imaginary part of the complex index represents absorption. The term critical angle, used for convenience in the above narrative, is anachronistic: it apparently dates from 1873. In the 20th century, quantum electrodynamics reinterpreted the amplitude of an electromagnetic wave in terms of the probability of finding a photon. In this framework, partial transmission and frustrated TIR concern the probability of a photon crossing a boundary, and attenuated total reflectance concerns the probability of a photon being absorbed on the other side. Research into the more subtle aspects of the phase shift in TIR, including the Goos–Hänchen and Imbert–Fedorov effects and their quantum interpretations, has continued into the 21st century. Gallery See also Notes References Bibliography S. Bochner (June 1963), "The significance of some basic mathematical conceptions for physics", Isis, 54 (2): 179–205; . M. Born and E. Wolf, 1970, Principles of Optics, 4th Ed., Oxford: Pergamon Press. C.B. Boyer, 1959, The Rainbow: From Myth to Mathematics, New York: Thomas Yoseloff. J.Z. Buchwald (December 1980), "Experimental investigations of double refraction from Huygens to Malus", Archive for History of Exact Sciences, 21 (4): 311–373. J.Z. Buchwald, 1989, The Rise of the Wave Theory of Light: Optical Theory and Experiment in the Early Nineteenth Century, University of Chicago Press, . O. Darrigol, 2012, A History of Optics: From Greek Antiquity to the Nineteenth Century, Oxford, . R. Fitzpatrick, 2013, Oscillations and Waves: An Introduction, Boca Raton, FL: CRC Press, . R. Fitzpatrick, 2013a, "Total Internal Reflection", University of Texas at Austin, accessed 14 March 2018. A. Fresnel, 1866 (ed. H. de Senarmont, E. Verdet, and L.