Total reflection

When light crosses a boundary between materials with different refractive indices, the light is partially refracted at the boundary and partially reflected. The total reflection is a phenomenon that happens when a ray of light strakes the medium boundary at an angle larger than a particular critical angle with respect to the normal to the surface.

The figure and program in section Refraction illustrates Snell’s law from the QED view angel. Snell (1591-1626) has derived his law on base of the wave theory. It relates indices n of the refractive index to the directions of propagation in terms of angles to the normal to boundary of two mediums: nαsinα = nβsinβ. In this formula, nβ and β are related to an optically slow medium (glass, water) and nα with α to the fast one (vacuum, air). The refractive index for vacuum equals to 1 by definition and to 1,00029 for air. For both is relevant the simplified equation sinα = nβsinβ.

Incidence angle β for which α = 900 is labelled as the critical angel of the substance. For water n = 1,33 and the critical angle is about β = 480. For the some glasses n = 1,55 and β = 420. The Snell’s law sinα = nsinβ has no solution for an incidence angle β greater than the critical one. The boundary in this case behaves as a mirror and rays are reflected according to the law: the angle of incidence equals to the angle of reflection.

Program reflection-total.exe simulates the refraction, reflection and the total reflection for substances water/glass on one side and air (vacuum) on the other side (Fig. 14) of the boundary. These phenomena are visualized by an interference pattern on three line detectors: the vertical Detector V, horizontal Detector H in air and horizontal Detector S in water/glass. The parallel paths of photon in water/glass are incident on boundary at an angle β to normal. In other words, the boundary is illuminated by a plane wave of monochromatic light. The white line shows some light rays as they are calculated according to the geometric optics laws.

Fig. 14 a) refraction rays, b) reflection for critical angle b, c) total reflection (reflection-total.exe)

Fig. 14a shows the situation where the incident angle b is smaller then the critical one. The photon paths in air are refracted and the photon causes an interference pattern on the vertical Detector V and the horizontal Detector H. Photon has also paths towards the Detector S in sub-stance, which results in interference pattern on this detector. It demonstrates that some part of light is reflected back to water.

Fig. 14b shows the situation where an incident angle b is the critical one. Some part of paths is directed to the vertical Detector V and photon creates here an interference pattern. Other part of paths has directions toward the Detector S and photon creates here an interference pattern as well. If some personification is allowed, we can say that a photon does not know whether it should continue its travel in the air or in the water.

Fig. 14c shows the situation where an incident angle b is greater then the critical one. The computation along the Detector H gives no interference pattern. This time only paths directed backward into water enable the photon to create an interference pattern on the Detector S and the Detector V. The incident light is totally reflected. This is the phenomenon, which makes the essence of several optical appliances; let us name optical fibers, which are used in endoscopes and telecommunications.

At last let us say, that all interference patterns are computed solely by the sum-over-paths method without using the Snell’s or reflection laws. Conversely, the interference patterns confirm the truth of these laws of geometric optics.

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