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
analogy^{1}^{}.
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 A_{1}B_{1}
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 A_{2}B_{2},
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:

OA_{2} / OB_{2} = v_{d} / v_{i }.
(Equation 1)

This situation lasts until the left wheel reaches the grass (configuration
A_{3}B_{3}). After that, the axle rolls in a
straight line (for example in configuration A_{4}B_{4}).

Analyzing the geometrical path of the axle, one can conclude that:

OA_{2} sin q_{i} = OB_{3}
sin q_{d}. (Equation 2)

With OB_{2} = OB_{3}, 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:

- the materion generally doesn't reach the boundary line perpendicularly to its velocity,
- the materion generally rotates as it reaches the boundary line,
- 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 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.

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.

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.

- 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. - Louis de Broglie developed an unappreciated pilot-wave theory (cf. his "Nouvelles perspectives en microphysique").