# Light

By

Ervin Chia

,

Chua Wei Hong

,

Hillson Hung

,

Nemo Chen

●

Published 2022-08-15

The study of light can be traced back 2500 years. People attempted to study and understand how light came to be: it was often the case that the Sun was the only natural source of light and thus revered then. The earliest known work on light was Optica by Euclid, studying the nature of light and how it behaved both qualitatively and quantitatively. Over the years, our understanding of light has vastly changed, especially so during the Scientific Revolution. Models of light have come and gone but most remain in one way or another due to its relevance in nature or practicality.

The nature of light was a contentious issue before the 1700s. The two leading theories of light's nature were particle-like (or corpuscular) and wave-like. In 1690, Christiaan Huygens published a work that would set in motion the victory of the wave theory, the Traité de la Lumière ¹. Summarily, his treatise put forth that light was best described by waves. This was primarily because of the following three phenomenon unique to transverse waves:

1. Diffraction
2. Refraction
3. Polarization

Light was found to be capable of exhibiting all three properties and Huygens mathematically proved the first two so. Although not widely accepted during its time of publication, contributions by several others such as Augustin-Jean Fresnel, Denis Poisson and Léon Foucault expanded on Huygen's work or proved the corpuscular theory otherwise, leading to the eventual acceptance of light as a wave in the mid-1800s.

In 1862, James Clerk Maxwell brought together several empirical laws that described electric fields \mathbf{E} and magnetic field \mathbf{B} and compiled them into a set of equations known (today) as the Maxwell's equations. By further combining the equations, he found that the \mathbf{E} and \mathbf{B} fields in vacuum obey

\begin{align}
\frac{\partial^{2}\mathbf{E}}{\partial x^{2}}+\frac{\partial^{2}\mathbf{E}}{\partial y^{2}}+\frac{\partial^{2}\mathbf{E}}{\partial z^{2}} & =\mu_{0}\epsilon_{0}\frac{\partial^{2}\mathbf{E}}{\partial t^{2}}\\
\frac{\partial^{2}\mathbf{B}}{\partial x^{2}}+\frac{\partial^{2}\mathbf{B}}{\partial y^{2}}+\frac{\partial^{2}\mathbf{B}}{\partial z^{2}} & =\mu_{0}\epsilon_{0}\frac{\partial^{2}\mathbf{B}}{\partial t^{2}}
\end{align}

Equations that look like the above are well studied: they are known as wave equations because solutions to this class of equations are of the form of waves. Through theory, he found that electromagnetic waves travelled at the speed of light, which had been fairly accurately measured by then. Additionally, treating light as an electromagnetic phenomenon would also solve the final piece of the puzzle; polarization of light. However, light was seen as entirely unrelated to electromagnetism then; thus this finding had stirred the whole scientific community as either proof of a new revolutionary theory or an amazing coincidence in nature.

Heinrich Hertz, in 1882, experimentally generated electromagnetic waves and showed that they travelled through space at the speed of light. Essentially he created the first radio 'antenna' and 'generator'. His work proved Maxwell's equations, which were not widely accepted then, as well as unified the study of light, electricity and magnetism into the same concept. Funnily enough, when asked about the practical importance of radio waves,

and when asked for potential uses of his antenna contraption, he said

We'll leave it at that, I guess.

Intuitively, a periodic wave can be described with three properties: frequency f, wavelength \lambda and its speed, v. The wavelength of a wave tells us how far, physically, does a wave extend before it repeats again. The frequency tells us how rapidly, in time, does a wave repeat itself. Finally, these two properties are related to the speed at which the wave travels by

\begin{align}
v=f\lambda
\end{align}

In the case of light, v is a famously known value: the speed of light. This quantity is conventionally denoted as c, and has the value c=299792458\text{ ms}^{-1}. Since the speed of light does not change at all (bar some circumstances), light can be entirely (and is often the case) described with only one of the two variables; application of equation 3 would govern the other. The wave-like nature of light motivates us to mathematically describe light in the form of a wave. In its most general form, the equation of a periodic wave looks like:

\begin{align}
y=A\sin(kx+\omega t+\phi)
\end{align}

where A,k,\phi\text{ and }\omega are constants that influence the way the function behaves. As a result, investigating light and its properties translates, mathematically, to solving for these 4 variables. x represents position in the x-axis and t represents time, which are typically the variables which we cannot control. In nomenclature, A is known as the amplitude, \phi the phase of the wave, k the angular wavenumber and \omega the angular frequency. k and \omega are related to wavelength and frequency via the following equations;

\begin{align}
k & =2\pi/\lambda\\
\omega & =2\pi f
\end{align}

Optics is the study of behaviour and properties of light and its interactions. We must therefore venture into the field of optics to understand how to measure the properties of light.

In earlier education, you may have come across a simple treatise in optics. Light is treated as rays, and lenses serve to bend the path of light. From this, we are able to predict how light would travel. The assumption that light can be treated as simple linear rays is known as the paraxial ray theory, and hinges on the key assumption in its namesake:

In any optical system that we study, we must first define a principal axis. The principal axis is an imaginary line that is drawn, such that if light would approach the system nearly parallel to this axis, it would interact the same regardless of position around the principal axis. This is known as having some degree of rotational symmetry; were you to rotate around the optical axis, the system would look the same to you. If one approaches the lens from above the optical axis, it must be affected in the same way as it would if it approaches from below the optical axis; in this case, they both converge to the same spot F in the diagram.

As much as we would like to treat light as rays for their simplicity, ray optics is extremely limited in explaining what light can do. If we wish to study the properties of light, then we must rely on a theory or method that hinges on the behaviour of light. The paraxial theory makes no concession for its wavelength; it is after all one of the earliest models of light. The ideas behind Huygen's Treatise are discussed from here on. To measure the wavelength of light, we must accept the behaviour of light as a wave, and as such treat it accordingly. Diffraction is one such phenomenon that is related to waves.

To understand diffraction, let us first take a step back to visualize a water wave. Imagine you throw a stone into a pond of water. What will you see?

We should see something that looks like the picture on the left, but intuitively we know that the pattern is made up of water waves that looks like the picture on the right.

The lines in picture on the left is what we call circular wavefronts. They are formed by (water) waves coming from the point source (your stone). Each line/wavefront represent a set of points at which the waves are in-phase (same \phi angle in the sinusoidal wave). If the light source comes from very far, the circularity of the wavefronts begin to become insignificant and instead appear parallel.

Diffraction is a phenomenon which occurs when waves meets an obstacle in the form of aperture (hole) or edge. The Huygens-Fresnel principle states that every point on a wavefront can be treated as a source of a circular wave. When the waves travel parallel, each point on a wavefront emits its own circular wavefront, and cancel out because of superposition (discussed right after this.), remaining parallel. However, if the parallel wavefronts are blocked by any opaque material, some sections of a parallel wavefront can no longer cancel each other out. As such one can have a diffraction phenomenon as such:

The superposition principle tells us that when waves physically overlap, their amplitudes can be added up. As such, the phases of the waves are important when superposing them; it can determine the amplitude of the individual waves at that point.

When two waves are in phase, such that their phase differences are even multiples of \pi, then they are said to constructively interfere and will add up. On the other hand, if they are odd integer multiples of \pi, then they destructively interfere with each other and results in a minimum. It is from the superposition principle that the Huygens-Fresnel principle explains and accounts for diffraction.

Go to Activity 1.

Since every gap can allow for the wave to act as a point source, then a series of gaps can be treated as a series of point sources at different fixed positions. This sets up a very interesting situation, as we are in position to consider what happens between the path differences between two sources. This is known as the double-slit diffraction problem, and is easily extendable to many arbitrary slits, or equivalent sources.

Light from two (or more) slits travel a different distance to the same point on a screen. Due to the path difference, there will be phase differences and constructive/destructive interference will occur. The condition for maximum constructive interference is

d\sin\theta=m\lambda

where m is an integer. As a result, we see alternating bright and dark spots on the screen as the angle \theta varies, due to constructive and destructive interference respectively.

We arrive at a phenomenon that depends on the wavelength of the light that we are observing: the maxima at which the interference pattern occurs depends on the wavelength. Light of different wavelengths will have maxima at different distances from the central maxima, which is denoted by y here. It was also found that the more slits there are, the sharper and clearer these maxima become. Thus, we employ diffraction gratings which have many grooves or gaps to allow light through, or reflect them, periodically across very small distances. Diffraction gratings can arise from (regular) imperfections on a surface; a metal ruler or even your phone screen can serve as a diffraction grating.

Go to Activity 2.

Occurring in the early 1900s was the ultraviolet catastrophe, an observation that was irreconcilable with the wave theory of light. Max Planck proposed a solution to this problem by unwillingly assuming that energy carried by light had to be quantized. The quantization of light was so successful in solving the problem that it even accurately predicted the entire emission spectrum of objects. This problem will be revisited later.

The success of quantization and particle-like light as a concept did not stop there. In 1905, Albert Einstein’s Nobel prize winning work on the photoelectric effect was also evidence of light quantization. In his experiment, he showed that by shining light of low frequency, no matter how intense, could not generate electrical current from a polished metal surface, indicating an insufficiency of energy to do so. However, once the frequency was raised past a certain value, an electrical current was observed, regardless of how strong the light was. This pointed towards some form of quantization with regards to the frequency (or equivalently, wavelength) of light. These successes reinvigorated the corpuscular theory of light.

In modern physics, we also treat light as quantized particles, capable of many things a classical particle could do such as carry momentum. Today, we understand that in the quantum regime, light exhibits both particle and wave nature, famously known as the wave-particle duality. The Planck-Einstein concept of light as a particle states that a single light particle known as the photon, carries a packet of energy E equivalent to

\begin{align}
E=hf=h\frac{c}{\lambda}
\end{align}

Where h is the Planck's constant, 6.626\times10^{-34}\text{ Js}^{-1} and the last equality holds from substitution using equation 3. This extended further meaning of a wave's properties into its particulate nature: the frequency of light not only denotes the rate at which it repeats, but also the energy it carries.

Go to Activity 3.