The text has been edited and the diagrams redrawn to fit a changed format but the essential physical argument is from Alexander Fishman - Professor of Physics at the University of Kazan (Russia).
Exactly how does light propagate in a medium, and exactly why is a partial reflection generated at a boundary between two transparent media of different refractive index?
The electric field of an incident light wave excites oscillations
of electrons in the medium. These oscillating electrons become
secondary sources, radiating spherical waves in all directions.
These waves, according to Huygens' principle, interfere, and form
the light field inside the medium.
Let us imagine that a plane monochromatic wave propagates in a
transparent medium (Figure
1). Consider the plane MN. The normal to the plane makes an
angle q with the incident light ray.
The electrons of the medium in this plane begin to oscillate.
Electrons in the small volume V1 radiate a secondary wave in all directions. Consider Ray 1, at an angle y to the normal. For every element V1, another element, V2, can be found on the plane MN such that V2 radiates a secondary wave in the same direction with the same amplitude, with a phase lag of p radians. The path difference between waves 1 and 2 is one half wavelength, l/2.
So, the distance d . between elements V1 and V2 is given by....
For any element on the surface MN we can find a second "twin" element, so that waves from both elements will cancel in all directions, with the exception of y = q (reflected wave) and y = (p - q) (rectilinear propagation).
Figure 1a shows both the reflection and rectilinear propagation. Now we will show that the reflected wave does not arise in the body of the medium.
If the medium is homogeneous we can always find an element V3 (Figure 2), which is separated from V1 in the direction of the incident wave by a distance d, such that the secondary waves 1 and 3 will have a phase lag of p radians. The path difference between these waves 1 and 3 must equal l/2. The distance d .between elements V1 and V3 is given by........
At once ...
When the separation of the elements V1 and V3 is l/(4cos2q) the secondary waves 1 and 3 will have a phase difference of p radians and the waves will cancel. A reflected wave for which y = q is therefore "forbidden" in a homogeneous medium. Only one propagating wave remains, for which y = (p - q).
Note: it is essential for our explanation, that all elements are exactly equivalent to each other. The medium must be completely homogeneous.
A reflected wave for which y = q is "forbidden" in a homogeneous medium, but can arise at the boundary between to different media for the following reason.
Consider the boundary between two different media (Figure 3). Now the elements V1 and V3 are located in different media with a different number of different atoms. As a result the elements V1 and V3 will radiate waves with different amplitudes. These waves are out of phase and will not cancel each other completely. A partial reflection arises at the boundary between the two different media.