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Summer has arrived, and you have been served a gorgeous-looking cocktail. Condensation droplets on the glass reveal you are set for a much needed particularly refreshing indulgence. However, just as you were about to soak up the colourful fluid of blissful gratification, your shockingly intelligent child asks why the straw seems to be broken inside your drink. And if not about that, then it’s about why this bear’s head is in the wrong place. Sure, the answer is light refraction, but why, exactly, do glass and liquids refract light?

We will provide you with the answer. However, before we begin, we need to ask, TL;DR? Rather not see formulas? Scroll down to the last section, the Quick summary. More curious? Then by all means, read on. I promise, not a single calculation will be done. And if you do read on, you will know actual physics. Shockingly more than most.

For your convenience, here’s a little table of contents:
A few incorrect explanations
Why are they incorrect?
Step 1. Not particles, not waves: it’s all fields
Step 2. Maxwell’s field equations
Step 3. Draw the vectors
Step 4. The electric field inside of materials
Quick summary: why, exactly, do glass and liquids refract light?

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A few incorrect explanations

What would you answer? Here are just a few bad examples which other people (but not you) tend to tell their offspring.

1. Light takes the fastest route. As its speed differs per material, it needs to change direction. Or: light takes the path of the least amount of action. Same reasoning.
2. When light enters the glass and the liquid, it bounces back and forth between the molecules and atoms of the material. Due to their crystalline or liquid arrangement, the overall direction of light changes. Hence, light is refracted.
3. Light consists of particles, so-called photons, which get absorbed by the atoms of the material, causing their electrons to temporarily increase their orbital radius around the nucleus. The instant they fall back to their original orbit, they emit another photon in a direction which depends on and is consistent with the type of atom, i.e. material. The overall result is that the beam of particles has changed direction. Hence, light is refracted.
4. Huygens’ Principle. Light is a wave. Every point on its wavefront can be a source for a circular wavelet. Draw them, connect the dots and you’ll see: light gets refracted.

Why are they incorrect?

1. Okay, this is not incorrect, however, while light does that, it doesn’t explain what really happens. It’s an answer to a different kind of question. So, to be ‘that person’ here, in terms of answer-to-the-question-asked, it’s incorrect after all.
2. By this logic, light should appear much more spread out due to the probabilistic nature of the supposed bouncing back and forth between chaotically moving or vibrating molecules and atoms. The specific direction of bouncing light is not guaranteed to be as consistent as we nevertheless observe in the real world. The resulting image should be a blur. It is not.
3. Here too, light should appear much more spread out. The direction of the re-released photon is not guaranteed to be in the direction we observe in the real word. A photon could be re-emitted in any direction, regardless of the type of atom. The frequency of the photon correlates with the atomic configuration, not its direction. Moreover, ‘getting absorbed’ and ‘re-emitted’ are not well defined. What does that even mean?
4. This is a sophisticated one. At first glance, it does produce an angle for the outbound light beam. However, Huygens’ Principle only corresponds to observations if you cherry pick from multiple possibilities. See Figure 1 for a brief explanation.

Step 1. Not particles, not waves: it’s all fields

So, what does make light refract then? We need to take a few mental steps. Here is the first one, which you’ll just have to get used to.

Space throughout the entire observable Universe is filled with fields. In fact, fields are a property of space. Space without fields does not exist. With space come fields. Points in most fields not only have a value, they also have a direction. They are called vector fields. Some are called scalar fields; their points have no direction, they only have values. There are more types of fields, such as tensor fields and fermionic fields. This is quantum field theory (QFT), the most successful and accurate theory to date. Has been for well over ninety years (including a renaissance in the 1970s).

Next question is, what concrete fields are we talking about? You probably heard of or read about the Higgs field¹. In 2012, the Large Hadron Collider at CERN produced an oscillation in the Higgs field or rather an excitation. That excitation is what we call the Higgs particle. The energy produced inside the LHC was more than enough to cause an excitation of the Higgs field, which we perceive as a particle². The field was proven to be a real thing. Two Nobel Prizes were awarded to François Englert and Peter Higgs for having proposed the existence of the Higgs field forty-eight years earlier. It proved how humans with their shockingly tiny brains were able to probe the depths of the subatomic world, a thousand times smaller than the atomic nucleus, and the entire observable Universe at the same time. By using maths and, forty-eight years later, by building ingenious experiments.

There are more fields. There is an electron field. Most of the time, the field has value zero. But when the values of a tiny part of that field oscillate at a distinct frequency, we call that an electron.

There is also an electromagnetic field. A stream of billions of local oscillations of a range of frequencies is what we call a beam of visible light. It’s practical to sometimes talk about it as it being particles (called photons) as well as it being waves (electromagnetic radiation). It depends on what you’re calculating.

You could say there is also a proton field, although there are more fundamental fields than this, for instance the quark and gluon fields (protons aren’t elementary particles, they consist of quarks and gluons). However, for the purpose of this post, we will work with the simpler notion of a proton field. A fairly local oscillation is a proton, which we usually perceive as a particle.

There are many more fields but to discuss them all would justify a separate article. Or several books. And a couple of years of study.

So, what is light, what are electrons, what are protons or quarks? Are they both particle and wave? No, that’s an old and misleading question. Are they sometimes particles, sometimes waves then? No, also not that.

‘Particles’ aren’t actual particles like tiny silver ball bearings or something like that. They are best described as a mathematical function, which we call the wave function (denoted by the symbol Ψ). When measured they are fairly local oscillations or excitations at specific frequencies in fields pervading through all of space, almost behaving like particles. Sometimes, it’s practical to mathematically model them as either particles or waves, depending on the situation. However, it’s meaningless to state they are either or both at the same time. It’s more accurate to just treat them as mathematical wave functions instead of anything elseA personal conviction is currently that the wave function is all there is. Elementary ‘particles’ are wave functions. Nothing more, nothing less. Favouring ‘tangible objects’ over ‘mere mathematical descriptions’, which, nevertheless have been proven to be incredibly accurate after billions and billions of experimental runs, is really just exposing our limited understand of quantum physics as confined by everyday, large-scale experiences such as playing with base-, basket- and footballs. There is no reason, however, to assume the latter are a measure to gauge the subatomic foundation of our Universe. In my view, that should be the wave function. — KJ.

Step 2. Maxwell’s field equations

James Clerk Maxwell was, besides Scottish, a scientist in the field of mathematical physics. Having studied the previous work of Faraday, Gauss, and Ampère, he showed that an electric field and a magnetic field were the same thing, just different aspects of it. That thing is what we now call the aforementioned, space-filling electromagnetic field. He also showed that light was an electromagnetic phenomenon. A disturbance in the field.

He formulated a set of four differential equations. This set bears his name. We will only ‘use’ two of the four:

\begin{align}
\mathbf{\nabla} \cdot \mathbf{E} &= \frac{\rho}{\varepsilon_0}, \\
\mathbf{\nabla} \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t}.
\end{align}

I say, ‘use’, but don’t worry, we’re not going to do any complicated calculations.

Equation (1) is called Gauss’s law and shows how the electric field (which is one aspect of the electromagnetic field), denoted by E, is influenced by a charge $\rho$, such as the negative charge of an electron or the positive charge of a proton. The symbol $\varepsilon_0$ denotes a constant, which differs depending on the material. The subscript 0 denotes it’s the constant of the vacuum of space. This physical constant $\varepsilon_0$ has different names such as vacuum permittivity, permittivity of free space or the electric constant. In this article, we will use different values for this constant, however. We will use

\[ \varepsilon_\text{air} \text{ and } \varepsilon_\text{mat}. \]

‘Mat’ is short for ‘material’ which could be glass or liquid, for example. The precise numerical values we won’t use, because that’s not important for understanding why light bends. These two epsilons will turn out to play a pivotal role in the bending of light by materials, however. Do read on, I’d say.

In Figure 3, the three fields in space are again depicted. This time, you can see the elevated values as blobs in the proton field. They are protons. They are surrounded by electron blobs as is depicted in the electron field. Both ‘particles’ influence the electromagnetic field, or, rather, the electric subfield thereof. Note that the blobs in the latter do not constitute ‘particles’, merely influences in the electric field.

Step 3. Draw the vectors

Have a look at Figure 4. The orange arrow or vector denotes the direction of the light inside whatever material we have, such as glass or a liquid. Maxwell showed that light, being an oscillation in the electromagnetic field, has an oscillatory component in the electric subfield of the electromagnetic field. And that electric field is orientated perpendicular to the direction of light. This is represented by the green vector.

It is important to note that the electric field vector has two fundamental components, namely a component vector parallel to the surface, denoted by the symbol $\parallel$, and a component vector perpendicular to the surface, denoted by the symbol $\perp$. This is depicted in Figure 5.

Exactly at the surface, the transition from the material to air, the electric field of the material and the electric field of the air will have to ‘slide’ to an equal value (or else we would have a tear in our universe). This means that

\begin{align}
\varepsilon_\text{air}(\mathbf{\nabla} \cdot \mathbf{E}_\text{air}) &= \varepsilon_\text{mat}(\mathbf{\nabla} \cdot \mathbf{E}_\text{mat}), \\
\mathbf{\nabla} \times \mathbf{E}_\text{air} &= \mathbf{\nabla} \times \mathbf{E}_\text{mat}.
\end{align}

If we do a little bit of calculus, we come to the following equations:

\begin{align}
\mathbf{E}_{\text{mat}\parallel} &= \mathbf{E}_{\text{air}\parallel} \\
\varepsilon_\text{mat}\mathbf{E}_{\text{mat}\perp} &= \varepsilon_\text{air}\mathbf{E}_{\text{air}\perp}.
\end{align}

So, the vector components of both air and the material parallel to the surface are equal. However, the vector components perpendicular to the surface are not. This is the crux:

\[ \varepsilon_\text{mat} \neq \varepsilon_\text{air}. \]

In fact,

\[ \varepsilon_\text{mat} > \varepsilon_\text{air}. \]

This means that the perpendicular vector component of air has to be larger than that of the material in order to satisfy equation (6). This is depicted in Figure 6.

The only thing left to do, is to draw the new direction of the light in the air. As we know that the direction of the electric field is perpendicular to the direction of light, courtesy to Maxwell and colleagues, we can easily construct the new course of the light outside in the open air as is depicted in Figure 7.

Step 4. The electric field inside of materials

The question is now: what causes the electric constant of materials to be so different?

Figure 8 shows a schematic depiction of what light, being the cause for disturbances in the electric field itself, does to electrically charged ‘particles’ inside a material in terms of quantum field perturbations. Note how the alignment of charges and thus the oscillations in the electromagnetic field have changed in such a way that the electric subfield as a whole, inside of the material, has to have changed values as well. These changes are encapsulated in the electric constant, $\varepsilon_\text{mat}$.

Light changes a material’s electromagnetic configuration, which then influences the trajectory of that same light.

Quick summary: why, exactly, do glass and liquids refract light?

The Universe, i.e. space itself³, contains an omnipresent electromagnetic field. Mathematically, we can divide this field up into two components: its electric (sub)field and its magnetic (sub)field.

Electrons in glass and liquids as well as light are influenced by the electric field. At the same time, they influence that same electric field. When light hits the material, it changes the electric field inside the material. This makes electrons bring about opposite electric field changes in turn. The net electric field inside the glass changes the light’s direction of propagation in a perfectly predictable way. Courtesy of vector calculus.

Or, if you prefer:

Light pushes on electrons via the electric field. Electrons push back a bit via the same field. Light says, ‘Okay, okay, relax!’, and takes a slightly different route.

1. It just so happens this is not a vector field; its points have no direction, just values, and so, it is a scalar field.[↩]
2. The Higgs particle itself was indirectly observed as its lifespan is too short. It decays quickly into other particles, or excitations, in other fields. Those, however, live long enough for the detectors to observe.[↩]
3. With ‘space’, we don’t mean ‘outer space’ but rather the thing we move in, the volume, the expanse, the invisible yet essential thing allowing us to move back-and-forth, up-and-down, left-and-right.[↩]