A single photon refracts

Arjen Dijksman © 2000

Light models that explain why a single light-particle (photon) refracts are scarce. There is the newtonian interface acceleration and its mechanical analogy of the marble that rolls on the horizontal backcover of an open book (figure 1). In this model the photon must have a higher velocity in water than in air, which doesn't fit with experimental values. There is also the Fermat least time analogy of a particle that "knows" which path it must take to arrive earlier at its target, like a dog on the shore that saves its master from drowning.

In the following, I present a deterministic photon model that fits with refraction. Let me introduce it with the simplified car analogy¹. As all car drivers know, when a wheel bites on the grass verge of the road, the car is pulled towards that side. If we see a car as an homogeneously powered axle with two solidly fixed wheels, it will have an incident velocity v_i on the asphalted road and a deviated velocity v_d on the grass surface. The friction on the grass is greater than the friction on the asphalt, therefore in this case we have v_i > v_d .
Figure 2 gives a geometrical interpretation of the phenomenon. The axle AB departs from the configuration A₁B₁ with a velocity v_i. This velocity forms an angle q_i with the normal to the boundary line. When the axle is in the configuration A₂B₂, the right wheel is constrained to roll at velocity v_d, while the other wheel still rolls at velocity v_i. Both wheels are constrained to draw an arc of a circle, because they must turn at the same rotational speed (remember they are solidly fixed to the axle). The centre O of the circle is determined by the relation:
OA₂ / OB₂ = v_d / v_i . (Equation 1)
This situation lasts until the left wheel reaches the grass (configuration A₃B₃). After that, the axle rolls in a straight line (for example in configuration A₄B₄).

Analyzing the geometrical path of the axle, one can conclude that:
OA₂ sin q_i = OB₃ sin q_d. (Equation 2)
With OB₂ = OB₃, equations 2 and 3 give the Snell-Fermat law for refraction:
sin q_i / sin q_d = v_i / v_d .
As the centre O of the circle is at the side of the grass, the axle will always pass the separation line. That is not the case for a car that drives on the grass towards the asphalt. As shown on figure 3, for an incident angle greater than a critical angle q_c, where sinq_c = v_grass / v_asphalt, the car will never pass the boundary line.

The behavior of a refracting photon and of the car in this particular case are similar, so let me state more precisely my photon model. In this model, the photon is represented by a straight line's segment of constant length d, which I call a materion. The velocity of the materion is determined by its contact interactions with the surrounding particles. The higher the density of the surrounding particles, the lower the velocity in this environment will be. Because the materion has an extent d, it will behave as the above axle when it is at the boundary between two media. There are three major differences between a materion and this axle analogy:

1. the materion generally doesn't reach the boundary line perpendicularly to its velocity,
2. the materion generally rotates as it reaches the boundary line,
3. there are lots of other materions that surround the single materion.

These three particularities open wide fields of investigation. As an introduction, I will only give some thoughts.

The materion generally doesn't reach the boundary line perpendicularly to its velocity

The angle between the direction AB of the materion and the boundary line may theoretically take any value. Figure 4 shows some possible 2D-interaction configurations. Imagine an arrow shot towards the surface of a lake. For a given incident angle, the arrow is certainly not refracted at the same angle, depending on the proper direction of the arrow. This corroborates with the QED-explanation of refraction (phase addition).We can even imagine that the arrow bounces on the water when it reaches the surface nearly parallelly.

Of course, the directions of the incident velocity and of the materion are generally not in the same plane, which complicates the investigation.

The materion generally rotates as it reaches the boundary line

When you throw a baseball bat, it generally rotates. The same yields for the materion. This particle has a dual nature. It has a wavelength and a frequency and therefore shows also wave properties. The baseball bat you threw will generally rotate in a plane parallel to its velocity. This single materion "wave" is therefore longitudinal (along the direction of the velocity), transverse and polarized.
The rotating speed (and therefore the frequency) of the materion has an influence on the refraction angle. Blue light doesn't refract with the same angle as red light.

There are lots of other materions that surround the single materion

The surrounding materions interact with the single materion. Imagine a field of needles. Through percussion, a rotating needle "pilots"² another wave in a field of needles, whether these needles have the same frequency or not. This can give rise to interferences even if the "pilot" needle is the only one rotating at visible light frequency.

References

1. For the car analogy, see for example "Thinking Physics, Practical lessons in critical thinking" by L.C. Epstein, Insight Press, San Francisco, 1987. Curiously this single car analogy is used to explain the wave-like behaviour of light, although wavefronts involve multiple particles. A better analogy for the wave theory is therefore the marching band that reaches muddy ground or waves reaching shallow water.
2. Louis de Broglie developed an unappreciated pilot-wave theory (cf. his "Nouvelles perspectives en microphysique").