# Fundamental phenomena governing light transport

As we are learning (or reminding ourselves) more and more about the nature of the light the time has come to show how photons can behave when they interact with matter. Namely which phenomena are governing the transport of the light (and any type of waves in general) and how they affect our perception:

1. Reflection,
2. Interference,
3. Diffraction,
4. Refraction.

However, keep in mind that today I am not going to explain in detail why these phenomena happen at the atomic (ground) level. I would rather focus on practical implications and mathematical relations that you should take for granted. Otherwise this post would be unnecessarily long.

#### Propagation of the bunch of photons

If we have a point source of light, let’s say LED, in an isotropic medium (where there are no privileged directions) the wave propagates as if it was an expanding sphere. Have a look at the picture below:

The picture presents the red diode on the ground that emits light isotropically in all directions in the upper hemisphere. Since all directions in the upper hemisphere are equally probable and the speed of light in that medium is constant, photons emitted at the same point of time are lying on the same surface. In that case this is a sphere so we are talking about spherical wave. Note, that photons are in constant movement, so the sphere they create is expanding:

$r(t + \Delta t) - r(t) = c\Delta t > 0$.

On the other hand the energy that these photons carry is constant. It must stay constant, otherwise the law of energy preservation would be violated! Knowing that we can define the intensity of light as amount of energy per unit surface area:

$I(r) = \frac{E_f}{S} = \frac{nhf}{4\pi r^2}=\frac{const}{r^2}$

So, if you want to illuminate the object which is 1 meter away from your source of light to be illuminated with the same amount of photons as that which is 2 meters away you need 4 times less optical energy. This is known as the inverse square law.

Looking at the picture above you can also tell something more than that. Look how the curvature of the sphere changes with the increase of its radius. You can easily add much bigger spheres and at some point you will notice that locally the wave will start to be more and more flat. From that point we can talk about plane waves where photons of the same age lay in the same plane.

#### Huygens principle

In the XVII century there was a Dutch physicist that formulated the rule which can be expressed as follows:

Every point in space to which wave approaches becomes the new source of spherical wave with the same properties (frequency, phase).

This is a general rule that was created knowing the experimental results. In fact Huygens was unaware of the reasons why waves behave in that way – nobody at his times was. What matters is that this principle works out of the box. You can use it easily to geometrically explain the behaviour of waves (and light). But first lets show something very simple.

#### Plane wave vs Huygens principle

Ok, you can think now: ‘Ha, how is it going to work for planar waves‘. Just look at this picture:

If we have a plane wave that approached some place in space we can draw as many spheres we want with the same radius from every point on that surface. Going with the number towards infinity you will have a new planar surface that is tangent to all spheres.

# Reflection

Let’s consider a plane wave AC that is moving towards a barrier (e. g. floor) lying on the line AB at some angle of incidence $\alpha_{GDJ}$. The point A to which the wavefront (surface made of points in space where waves are in the same phase; in presented case wavefront is a simple plane) comes first becomes a new source of wave while the original wavefront still needs time $\Delta t$ to reach point B. So the spherical bubble will be growing during $\Delta t$ and will eventually reach radius AE equal to the DB section. If you draw the tangent EB to the sphere after $\Delta t$ you will get a reflected wavefront. That new wave will be propagating in the EF direction. The relationship between angles $\alpha_{GDJ}$ and $\beta_{JHD}$ is simple equality $\alpha_{GDJ} = \beta_{JHD}$:

Angle of incidence and reflection are equal.

The result is the same as in point mass mechanics (the problem of billiard player). Pretty simple, isn’t it?

One more thing is worth to mention. Angles of incidence and reflection are always measured relatively to the normal IJ. Normal is a section that is always perpendicular to the tangent which the wave hits.

# Interference

If at any point in space two waves meet the total (temporary) amplitude will be an algebraic sum of amplitudes of these two waves (of course there can be even more than 2 waves that can interfere).

For the moment start to think about yourself as of point in space. Every month you receive salary/scholarship of +1000 but you have to pay for your flat, food, clothes and others -1200. So you are starting to live on the credit (-200). Fortunately, you have your parents that helped you with +250. Your balance at the end of the month is +50.

In that short comparison you are the point in space at which waves of money meet and your amplitude is the final account balance. Physics is that simple.

Mathematically, interference comes from the linearity of the wave equation (you should remember it from the previous post):

$\frac{\partial^2 u(x, t)}{\partial x^2} = \frac{1}{c^2}\frac{\partial^2 u(x, t)}{\partial t^2}$

$\sum_i^n\frac{\partial^2 u_i(x, t)}{\partial x^2} = \frac{1}{c^2}\sum_i^n\frac{\partial^2 u_i(x, t)}{\partial t^2}$

Let’s look now at the simple example of two waves interference. Both of them have the same wavelength and amplitude but are propagating in the opposite directions:

From that animation you can tell that:

• Interference is dynamic (time and space dependant) phenomenon.
• There are points in space where total amplitude is two times bigger than the amplitude of the single wave.
• On the other hand there are points (nodes) that seem to be fixed. They have no evolution.

This particular arrangement when opposing waves meet is called a standing wave. It is just because there are fixed points that exhibit none (nodes) and maximum oscillations regardless of time. If we move to the acoustics and musical instruments like trumpet or flute standing waves are the core mechanism that rules their sound.

#### How does interference affect energy?

The total energy of waves that interfere is the sum of energies of particular waves $E_{tot} = \sum_i^n E_i$. Let’s stop for a moment and go back to the formula for the energy of the photon:

$E_i = hf_i$

If we have, let’s say, 5 photons of the same frequency $f$ the total energy will be $E_{tot}=5hf$. Does it mean that this created another wave with a frequency 5 times bigger than the initial one? No. We still have 5 individual photons. No new photon was created! Interference affects only intensity of the wave.

#### Summary

I think it might be beneficial if I cut this post at this point and leave description of refraction and diffraction for the next time. These two phenomena are that important for the camera optics that they just deserve the special treatment.

You might have not realized at first but energy preservation principle makes it very difficult to enlight distant objects. The fall off $\frac{1}{r^2}$ is very rapid.

Huygens principle is just a rule of a thumb that allows to neatly describe wave behavior. It says nothing about the real reasons of phenomena but it is very useful for our current needs. We will be using it even more extensively next time.

Reflection and interference might seem simple but combined may be the basis for very sophisticated experiments. One can mention LIGO and VIRGO – laser interferometers that are built to detect extremely small space displacements caused by gravitational waves (very sexy – trendy – topic nowadays).

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Paweł Jurgielewicz
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