Introduction

Historically, the nature of light has been a subject of scientific inquiry for centuries. In the 17th century, two prominent models emerged to explain the behavior of light.

In 1637, Descartes proposed a corpuscular model, treating light as a stream of particles. This model successfully explained fundamental phenomena like reflection and refraction. Notably, based on this model, Descartes predicted that light would travel faster in a denser medium (like water) when it bends towards the normal during refraction. Isaac Newton further championed and developed this particle theory in his work 'Opticks', leading to the model often being associated with him due to his influence.

However, in 1678, Christiaan Huygens presented an alternative: the wave theory of light. This model also provided satisfactory explanations for reflection and refraction. Crucially, the wave theory made a contradictory prediction regarding the speed of light upon refraction into a denser medium. According to Huygens' theory, if light bends towards the normal, its speed must be slower in the second medium. This prediction was later confirmed experimentally by Foucault in 1850, demonstrating that the speed of light in water is indeed less than in air, which supported the wave model over the simple corpuscular model.

Despite experimental evidence favoring the wave theory, its acceptance was initially slow. A primary reason was the prevalent belief that waves required a medium to propagate, posing a challenge for explaining light's travel through the vacuum of space. The wave nature of light gained significant ground with Thomas Young's groundbreaking interference experiment in 1801, which provided compelling evidence that light is a wave phenomenon. This experiment also allowed for the first measurements of the wavelength of visible light, found to be extremely small (on the order of 0.5 micrometers for yellow light).

The smallness of light's wavelength relative to common objects like mirrors and lenses allows us to treat light propagation as approximately along straight lines under many circumstances. This approximation forms the basis of geometrical optics, discussed in the previous chapter. In the limit where wavelength approaches zero, a ray is defined as the path of energy propagation.

Following Young's work, interference and diffraction experiments over the next few decades solidified the wave theory. The puzzle of light propagation through vacuum was resolved by James Clerk Maxwell in the mid-19th century. Based on his unified theory of electricity and magnetism, Maxwell predicted the existence of electromagnetic waves, which are self-propagating oscillations of electric and magnetic fields that do not require a material medium. He calculated the speed of these waves in vacuum and found it matched the measured speed of light, leading to the conclusion that light is an electromagnetic wave.

In this chapter, we will delve into Huygens' principle and use it to derive the laws of reflection and refraction. We will then explore the phenomena of interference, based on the principle of superposition, and diffraction, which relates to the bending of waves. Finally, we will discuss polarisation, a characteristic property of transverse waves like light.

Huygens Principle

Huygens' principle provides a geometrical method to determine the position of a wavefront at a future time, given its position at a previous instant. Before describing the principle, we must understand the concept of a wavefront.

A wavefront is defined as a surface (or line in 2D) connecting all points that are oscillating in the same phase. For instance, when a stone is dropped in water, circular ripples spread out. All points on one such circular crest or trough are in the same phase, forming a circular wavefront.

The wavefront moves outwards from the source at the speed of the wave. Importantly, the direction of energy propagation of the wave (which we visualize as rays in geometrical optics) is always perpendicular to the wavefront.

• For a point source emitting waves uniformly in all directions in a homogeneous medium, the wavefronts are spherical (
  Figure 10.1(a)).
• At a very large distance from a point source, a small segment of a spherical wavefront appears nearly flat. Such a wavefront is called a plane wave (
  Figure 10.1(b)).

Now, Huygens' principle states:

"Every point on a given wavefront may be considered as a source of secondary spherical wavelets, which spread out in all directions with the speed of the wave in that medium. The new position of the wavefront at any later time is the envelope (common tangent surface) of these secondary wavelets."

Let's illustrate with a spherical wavefront at time $t=0$ (

Figure 10.2). To find the wavefront at time $t$, we take several points on the initial wavefront (F$_1$F$_2$). From each point, we draw a spherical wavelet with a radius equal to $vt$, where $v$ is the wave speed. The surface that is tangent to all these wavelets in the forward direction (G$_1$G$_2$) represents the new wavefront at time $t$.

A limitation of this simple construction is that it also produces a backward-moving envelope (D$_1$D$_2$). Huygens addressed this by assuming the amplitude of secondary wavelets is zero in the backward direction and maximum in the forward direction. While this assumption works, a more rigorous wave theory is needed for a complete justification of the absence of the backwave.

Similarly, for a plane wave propagating in a medium (

Figure 10.3), each point on the initial plane wavefront acts as a source of secondary wavelets. The envelope of these wavelets after time $t$ will be another plane parallel to the original wavefront, representing the new position of the plane wavefront. The lines perpendicular to the wavefronts represent the rays.

Refraction And Reflection Of Plane Waves Using Huygens Principle

Refraction Of A Plane Wave

We can use Huygens' principle to derive Snell's law, which governs the bending of light as it passes from one medium to another. Consider a plane wavefront AB incident on the interface PP' separating medium 1 (with wave speed $v_1$) and medium 2 (with wave speed $v_2$) at an angle of incidence $i$ (

Figure 10.4). Let the wavefront reach the interface at point A first. As the wavefront propagates, point B reaches the interface at C after time $t$, so the distance BC = $v_1 t$.

During this same time $t$, the secondary wavelet originating from point A in medium 2 will have traveled a distance AE = $v_2 t$. According to Huygens' principle, the new refracted wavefront is the common tangent to all secondary wavelets in medium 2. If we draw a tangent from C to the sphere of radius $v_2 t$ centered at A, the tangent plane CE represents the refracted wavefront.

In the right-angled triangle ABC, $\sin i = \frac{\text{BC}}{\text{AC}} = \frac{v_1 t}{\text{AC}}$.

In the right-angled triangle AEC, the angle of refraction is $r$. So, $\sin r = \frac{\text{AE}}{\text{AC}} = \frac{v_2 t}{\text{AC}}$.

Dividing the two equations:

$$\frac{\sin i}{\sin r} = \frac{v_1 t / \text{AC}}{v_2 t / \text{AC}} = \frac{v_1}{v_2}$$

This is Snell's law in terms of wave speeds. If the ray bends towards the normal ($r < i$), then $\sin r < \sin i$, which implies $v_2 < v_1$. This confirms the wave theory's prediction that light slows down when entering an optically denser medium.

Using the definition of refractive index $n = c/v$ (where $c$ is the speed of light in vacuum), we have $v_1 = c/n_1$ and $v_2 = c/n_2$. Substituting these into the speed ratio:

$$\frac{\sin i}{\sin r} = \frac{c/n_1}{c/n_2} = \frac{n_2}{n_1}$$

Rearranging gives the familiar form of Snell's law: $n_1 \sin i = n_2 \sin r$.

The frequency of light ($f = v/\lambda$) remains constant when light travels from one medium to another, as the frequency is determined by the source. Thus, $v_1 = f \lambda_1$ and $v_2 = f \lambda_2$. Substituting into the speed ratio, $\frac{v_1}{v_2} = \frac{f \lambda_1}{f \lambda_2} = \frac{\lambda_1}{\lambda_2}$. Therefore, $\frac{\sin i}{\sin r} = \frac{\lambda_1}{\lambda_2}$. This means that when light enters a denser medium ($v_2 < v_1$), its wavelength also decreases ($\lambda_2 < \lambda_1$), while its frequency remains unchanged.

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Refraction At A Rarer Medium

If light travels from a denser medium (medium 1, speed $v_1$) to a rarer medium (medium 2, speed $v_2$), then $v_2 > v_1$. Following the same Huygens construction as above, the refracted wavefront bends away from the normal ($r > i$) (

Figure 10.5). Snell's law ($n_1 \sin i = n_2 \sin r$ or $\frac{\sin i}{\sin r} = \frac{v_2}{v_1}$) still holds.

In this case, since $v_2 > v_1$, the ratio $\frac{v_2}{v_1}$ is greater than 1. This implies $\sin r = \frac{v_2}{v_1} \sin i > \sin i$. As the angle of incidence $i$ increases, the angle of refraction $r$ increases even more. There is a maximum possible value for $\sin r$, which is 1 (when $r = 90^\circ$). The angle of incidence in the denser medium corresponding to an angle of refraction of 90° in the rarer medium is called the critical angle ($i_c$).

At $i = i_c$, $r = 90^\circ$. Applying Snell's law: $n_1 \sin i_c = n_2 \sin 90^\circ = n_2 \times 1$.

$$\sin i_c = \frac{n_2}{n_1}$$

where $n_1$ is the refractive index of the denser medium and $n_2$ is the refractive index of the rarer medium ($n_1 > n_2$). For any angle of incidence $i > i_c$, Snell's law would require $\sin r > 1$, which is impossible. Therefore, for angles of incidence greater than the critical angle, there is no refracted wave; all the light is reflected back into the denser medium. This phenomenon is called Total Internal Reflection (TIR).

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Reflection Of A Plane Wave By A Plane Surface

Huygens' principle can also be used to derive the laws of reflection. Consider a plane wavefront AB incident on a reflecting surface MN at an angle of incidence $i$ (

Figure 10.6). Let the wave speed in the medium be $v$. As the wavefront propagates, point B reaches the surface at C after time $t$, so BC = $vt$.

According to Huygens' principle, during this time $t$, a secondary wavelet originating from point A (which reached the surface at $t=0$) will have spread out in the same medium. The radius of this wavelet will be $vt$. The reflected wavefront is the common tangent to all such secondary wavelets. Drawing a tangent from C to the sphere of radius $vt$ centered at A, we get the reflected wavefront CE. Thus, AE = $vt$.

Now consider the triangles ABC and AEC. Both are right-angled triangles (at B and E respectively). They share a common hypotenuse AC. Also, BC = AE = $vt$. Therefore, triangles ABC and AEC are congruent by the RHS (Right angle-Hypotenuse-Side) congruence criterion.

Since the triangles are congruent, their corresponding angles are equal. The angle of incidence $i$ is the angle between the incident wavefront AB and the surface MN, which is equal to the angle between the incident ray (perpendicular to AB) and the normal to MN. Similarly, the angle of reflection $r$ is the angle between the reflected wavefront CE and the surface MN, which is equal to the angle between the reflected ray (perpendicular to CE) and the normal to MN.

From the congruence of triangles ABC and AEC, we have $\angle BAC = \angle ECA$. $\angle BAC$ is the angle of incidence $i$. $\angle ECA$ is the angle of reflection $r$. Thus, $i = r$. This is the first law of reflection (angle of incidence equals angle of reflection).

Furthermore, the incident ray (normal to AB), the reflected ray (normal to CE), and the normal to the surface MN all lie in the plane of the diagram, which demonstrates the second law of reflection.

Huygens' principle can also be used to explain how wavefronts change shape when interacting with optical components like prisms, lenses, and mirrors (

Figure 10.7). For instance, when a plane wave hits a convex lens, the central part passes through the thickest glass and is slowed down the most, causing the emerging wavefront to be curved inwards (spherical and converging). For a concave mirror, a plane wavefront reflects to become a spherical wavefront converging towards the focus.

An important consequence of Huygens' principle is that the time taken for light to travel from any point on an object wavefront to the corresponding point on the image wavefront is the same along any ray path. This concept is related to Fermat's Principle of Least Time.

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Coherent And Incoherent Addition Of Waves

The behavior of waves when they overlap or combine is described by the principle of superposition. According to this principle, when two or more waves meet at a point in a medium, the resultant displacement at that point is the vector sum of the individual displacements produced by each wave.

Consider two sources, S$_1$ and S$_2$, emitting waves. When these waves overlap, the resulting displacement at any point depends on the amplitude and phase of the waves from each source. If the sources are coherent, they maintain a constant phase difference between the waves they emit over time. This allows for a stable interference pattern.

Let's say two coherent sources S$_1$ and S$_2$ are vibrating in phase and producing waves with amplitude $a$. The displacement at a point P due to S$_1$ is $y_1 = a \cos(\omega t)$. The displacement due to S$_2$ is $y_2 = a \cos(\omega t + \phi)$, where $\phi$ is the phase difference at point P. This phase difference is related to the path difference $\Delta r = \text{S}_2\text{P} \sim \text{S}_1\text{P}$ by $\phi = (2\pi/\lambda) \Delta r$.

The resultant displacement at P is $y = y_1 + y_2 = a (\cos(\omega t) + \cos(\omega t + \phi))$. Using trigonometric identities, this simplifies to $y = 2a \cos(\phi/2) \cos(\omega t + \phi/2)$.

The amplitude of the resultant wave is $A_R = 2a |\cos(\phi/2)|$. Since intensity is proportional to the square of the amplitude ($I \propto A^2$), the intensity at point P is $I = 4I_0 \cos^2(\phi/2)$, where $I_0$ is the intensity produced by each source individually ($I_0 \propto a^2$).

• Constructive Interference: Occurs when the waves arrive in phase, resulting in maximum amplitude and intensity. This happens when the phase difference $\phi = 0, \pm 2\pi, \pm 4\pi, \dots$ (even multiples of $\pi$). Correspondingly, the path difference $\Delta r = 0, \pm \lambda, \pm 2\lambda, \dots$ (integer multiples of wavelength $n\lambda$, where $n$ is an integer). The maximum intensity is $I_{max} = 4I_0$.
• Destructive Interference: Occurs when the waves arrive out of phase, resulting in minimum (often zero) amplitude and intensity. This happens when the phase difference $\phi = \pm \pi, \pm 3\pi, \pm 5\pi, \dots$ (odd multiples of $\pi$). Correspondingly, the path difference $\Delta r = \pm \lambda/2, \pm 3\lambda/2, \pm 5\lambda/2, \dots$ ((n+1/2)$\lambda$, where $n$ is an integer). The minimum intensity is $I_{min} = 0$ (if amplitudes are equal).

If the two sources are incoherent, the phase difference between them changes randomly and rapidly over time. In this case, the interference term in the intensity formula ($4I_0 \cos^2(\phi/2)$) averages out to $2I_0 (\text{average of } \cos^2(\phi/2)) = 2I_0 \times (1/2) = I_0$. The resultant intensity at any point is simply the sum of the intensities from the individual sources:

$$I_{total} = I_1 + I_2 = I_0 + I_0 = 2I_0$$

With incoherent sources, there is no stable interference pattern of bright and dark regions. Instead, the intensity is uniformly distributed and is just the sum of the intensities from each source. This is typically what happens when two separate light bulbs illuminate a surface.

Figure 10.8 illustrates coherent sources (two needles) creating a stable interference pattern in water, with regions of maximum displacement (antinodes, A) and minimum displacement (nodes, N).

Interference Of Light Waves And Young’s Experiment

Observing interference with light from ordinary sources is difficult because light from such sources (like a sodium lamp) is emitted by individual atoms undergoing transitions. These emission events are independent and random, leading to rapid and unpredictable phase changes in the emitted waves (typically on the order of $10^{-10}$ seconds). Therefore, two independent light sources are usually incoherent, and their intensities simply add up, washing out any interference pattern (

Figure 10.11).

To demonstrate light interference, we need coherent light sources. Thomas Young achieved this in his famous 1801 double-slit experiment by deriving the waves from a single source. He used a single narrow pinhole S illuminated by a light source. The spherical wavefront spreading from S then illuminated two closely spaced pinholes, S$_1$ and S$_2$, on an opaque screen (

Figure 10.12(a)).

Since S$_1$ and S$_2$ are illuminated by the same wavefront from S, the light waves emanating from them are in phase (or have a constant phase difference determined by their positions relative to S). Any phase change in the original light from S is simultaneously reflected in the waves from S$_1$ and S$_2$, maintaining their coherence. Thus, S$_1$ and S$_2$ act as two coherent secondary sources.

When the waves from S$_1$ and S$_2$ overlap on a screen placed some distance away, they interfere, producing a stable interference pattern consisting of alternating bright and dark bands, known as fringes (

Figure 10.12(b), Figure 10.13).

Let the distance between the slits S$_1$ and S$_2$ be $d$, and the distance from the slits to the screen be $D$. Consider a point P on the screen at a distance $x$ from the center C. The path difference between the waves from S$_1$ and S$_2$ arriving at P is approximately $\Delta r = \text{S}_2\text{P} - \text{S}_1\text{P}$. For $D \gg d$ and $x \ll D$, this path difference can be approximated as:

$$\Delta r \approx \frac{xd}{D}$$

Based on the conditions for constructive and destructive interference from coherent sources:

• Bright fringes (Constructive interference): Occur when the path difference is an integer multiple of the wavelength $\lambda$.
  $\Delta r = n\lambda$, where $n = 0, \pm 1, \pm 2, \dots$
  So, the position of the $n$-th bright fringe is given by: $x_n = \frac{n\lambda D}{d}$
• Dark fringes (Destructive interference): Occur when the path difference is a half-integer multiple of the wavelength $\lambda$.
  $\Delta r = (n + 1/2)\lambda$, where $n = 0, \pm 1, \pm 2, \dots$
  So, the position of the $n$-th dark fringe is given by: $x_n = \frac{(n + 1/2)\lambda D}{d}$

The central fringe ($n=0$ for bright fringe) is at $x=0$, where the path difference is zero, and it is always bright. From these formulas, we can see that the positions of consecutive bright or dark fringes are equally spaced. The distance between two consecutive bright fringes (or two consecutive dark fringes) is called the fringe width ($\beta$):

$$\beta = \frac{\lambda D}{d}$$

Young's experiment was a crucial demonstration of the wave nature of light and provided a method to measure the wavelength of visible light.

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Diffraction

Diffraction is the phenomenon where waves bend or spread out as they pass through an opening or around an obstacle. It's essentially the deviation of waves from rectilinear propagation (traveling in straight lines) when they encounter boundaries. Like interference, diffraction is a characteristic property of all wave types, including light, sound, and water waves.

While we are accustomed to light casting sharp shadows (rectilinear propagation), this is an approximation that holds when the wavelength of light is much smaller than the size of the obstacles or apertures. When light encounters objects or openings comparable to or smaller than its wavelength, diffraction effects become noticeable. For example, the fine details visible on the edge of a shadow are due to diffraction, and the shimmering colours observed on a CD surface are also a result of diffraction by the closely spaced tracks.

Diffraction fundamentally limits the ability of optical instruments (like telescopes and microscopes) to resolve or distinguish between very closely spaced objects. The spreading of light due to diffraction prevents point sources from forming perfect point images.

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The Single Slit

Even a single narrow slit can produce an interference-like pattern, which is referred to as diffraction. When monochromatic light passes through a single narrow slit, instead of just producing a bright line corresponding to the geometrical image of the slit, a broader pattern is observed on a screen. This pattern consists of a wide, bright central band, flanked by alternating dark and progressively weaker bright bands on either side (

Figure 10.15).

This phenomenon can be understood using Huygens' principle. Each point within the single slit acts as a source of secondary wavelets. These wavelets interfere with each other, producing the observed diffraction pattern. Consider a plane wavefront incident normally on a slit of width $a$. We can imagine dividing the slit into a large number of very small segments, each acting as a coherent source of secondary wavelets since they are part of the same wavefront (

Figure 10.14).

The central point (C) on the screen receives wavelets from all parts of the slit with zero or negligible path difference, leading to constructive interference and the maximum intensity (the central bright fringe).

For points away from the center, there are path differences between wavelets originating from different parts of the slit. Dark fringes (minima of intensity) occur at angles $\theta$ where the path difference between wavelets from the edges or specific points in the slit leads to complete destructive interference. The condition for the first minimum occurs when the path difference between wavelets from the top edge and the bottom edge of the slit is $\lambda$. Dividing the slit into two halves, the wavelet from the top of the upper half cancels the wavelet from the top of the lower half (path difference $\lambda/2$), and so on across the slit. This occurs at an angle $\theta$ such that $\sin \theta = \lambda/a$.

In general, dark fringes (minima) occur at angles $\theta$ where $\sin \theta = \frac{n\lambda}{a}$, for $n = \pm 1, \pm 2, \pm 3, \dots$. Note that $n=0$ corresponds to the central maximum. Bright fringes (secondary maxima) occur roughly halfway between the dark fringes, at angles $\sin \theta \approx \frac{(n + 1/2)\lambda}{a}$, for $n = \pm 1, \pm 2, \pm 3, \dots$. The intensity of these secondary maxima decreases rapidly as $n$ increases.

The central maximum is significantly wider than the secondary maxima and contains most of the light energy. Its width is $2\lambda D/a$. The width of the secondary maxima is $\lambda D/a$.

The distinction between interference and diffraction has historically been debated. Richard Feynman commented that there's no fundamental physical difference; it's largely a matter of usage, with "interference" typically referring to the superposition of waves from a few sources (like two slits), and "diffraction" referring to the superposition of wavelets from a continuous distribution of sources (like the points within a single slit or around an obstacle).

In Young's double-slit experiment, the observed pattern is, in fact, a combination of the interference pattern from the two slits and the diffraction pattern from each individual slit. The overall pattern is an interference pattern modulated by the single-slit diffraction pattern.

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Seeing The Single Slit Diffraction Pattern

The single-slit diffraction pattern can be easily observed using simple household items. A narrow slit can be formed by holding two razor blades closely together with their edges parallel. When a clear glass electric bulb with a straight filament is viewed through this narrow slit, the filament acts as the light source. The eye's lens focuses the diffraction pattern formed by the slit onto the retina, where it can be observed (

Figure 10.16).

With careful adjustment of the slit width and parallelism, the bright and dark diffraction bands become visible. Using colored filters can make the pattern clearer by providing nearly monochromatic light. Observing with different filters (like red and blue) demonstrates that the width of the diffraction bands depends on the wavelength ($\beta \propto \lambda$), with red light producing wider fringes than blue light.

In both interference and diffraction, light energy is not created or destroyed but is redistributed. Dark regions occur where energy is minimum, and bright regions occur where it is maximum, adhering to the principle of conservation of energy.

Polarisation

Polarisation is a phenomenon that provides crucial evidence for the transverse nature of light waves. Consider a wave propagating along a string stretched horizontally. If the string is vibrated up and down, the displacement is perpendicular to the direction of propagation (transverse), and the vibration is confined to the vertical plane (say, the xy-plane if the string is along the x-axis and displacement is along the y-axis) (

Figure 10.17). This is a linearly polarised or plane polarised wave.

Similarly, if the vibration is side-to-side (in the xz-plane), it's another linearly polarised wave. For an unpolarised wave, the direction of vibration in the plane perpendicular to the propagation direction changes randomly and rapidly with time.

Light waves are transverse electromagnetic waves, meaning the electric field vector (and magnetic field vector) oscillates perpendicular to the direction of wave propagation. Ordinary light sources emit unpolarised light; the electric field oscillations occur randomly in all possible directions within the plane perpendicular to the ray.

A polaroid is a material that produces linearly polarised light from unpolarised light. It contains long-chain molecules aligned in a specific direction. When unpolarised light passes through a polaroid, the electric field components oscillating parallel to the direction of the aligned molecules are absorbed. The components oscillating perpendicular to this direction are transmitted. This preferred direction of transmission is called the pass-axis of the polaroid.

If unpolarised light of intensity $I_{unpol}$ passes through a polaroid, the transmitted light is linearly polarised along the pass-axis, and its intensity is reduced to half: $I_{pol} = I_{unpol}/2$. Rotating this single polaroid does not change the intensity of the transmitted beam as long as the incident light is unpolarised.

However, if this linearly polarised light then passes through a second polaroid (called the analyzer), the transmitted intensity depends on the relative orientation of the pass-axes of the two polaroids (

Figure 10.18). Let the first polaroid (polarizer) transmit light polarized with electric field amplitude $E_0$. If the pass-axis of the second polaroid makes an angle $\theta$ with the direction of polarization of the incident light, the component of the electric field transmitted by the second polaroid is $E = E_0 \cos \theta$.

Since intensity is proportional to the square of the amplitude ($I \propto E^2$), the transmitted intensity $I$ is related to the intensity of the polarized light incident on the second polaroid ($I_0 \propto E_0^2$) by:

$$I = I_0 \cos^2 \theta$$

This is known as Malus's Law. As the second polaroid is rotated, the transmitted intensity varies from maximum ($I_0$, when $\theta = 0^\circ$ or $180^\circ$, axes are parallel) to minimum (0, when $\theta = 90^\circ$ or $270^\circ$, axes are crossed).

Polaroids are used in various applications to control light intensity or reduce glare, such as in sunglasses, camera filters, and liquid crystal displays (LCDs). The phenomenon of polarisation is a definitive proof that light waves are transverse.

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