Refraction, diffraction and interference (3.3.2)

The principle of superposition states that when two or more waves meet at a point, the resulting displacement is the sum of the individual displacements from each wave at that point.

This can lead to constructive interference (where waves add up to create a larger amplitude) or destructive interference (where waves cancel each other out).

In resonance tubes and musical instruments like organ pipes, superposition creates standing or stationary waves.

When sound waves travel back and forth in a confined space (like a tube), they reflect at the ends and meet other waves travelling in the opposite direction.

This superposition causes the wave pattern to ‘stand still’ with specific points, called nodes (where there’s no movement or oscillations) and antinodes (where movement or oscillation is greatest).

In an organ pipe, blowing into it creates a sound wave that reflects off the closed end, producing a stationary wave pattern within the pipe. Each pipe length produces a unique pitch based on its stationary wave pattern.

In experiments, two speakers connected in stereo to the same frequency generated by a signal generator create sound waves that superpose in the air. By walking directly in front of the speakers, one can experience interference firsthand.

• Constructive Interference: When waves meet in phase, their amplitudes add up, resulting in louder sounds.
• Destructive Interference: When waves meet out of phase, their amplitudes cancel, resulting in softer sounds or silence.

This alternating pattern of loud and quiet regions is a clear indication of superposition, as sound intensity varies depending on your position relative to the speakers. This effect is used in acoustics and sound engineering to manage sound placement in concert halls and theatres.

The principle of superposition can also be demonstrated with diffraction gratings:

1. They have many slits close together, which enhances the interference pattern by producing bright and sharp lines.
2. Light diffracts through each slit, causing multiple waves to overlap and superpose.
3. The distance between bright fringes in the pattern depends on the wavelength of the light.

The principle of superposition can be demonstrated with Young’s double-slit experiment:

1. Light from a single source passes through two narrow slits.
2. The two waves emerging from these slits then overlap, leading to interference on a screen.
3. The overlapping light waves create bright regions (constructive interference) and dark regions (destructive interference), forming a pattern of fringes.
4. Bright fringes occur where the waves reinforce each other, while dark fringes occur where they cancel out.

The principle of superposition applies to all waves, not just sound and visible light. including the full electromagnetic spectrum.

An example of this is in a microwave oven, where reflected microwaves superpose to form standing waves with fixed high-energy (antinodes) and low-energy (nodes) points. Antinodes heat food most effectively. Rotation is essential for even cooking, as stationary food would have cold spots (nodes).

To create superposition in controlled experiments, two microwave transmitters are used, each emitting a wave that meets in the same region. A microwave detector can then identify areas of constructive interference (high signal strength) and destructive interference (low signal strength).

A stationary wave can also be created using a single transmitter and a reflector. The microwave reflects back towards the transmitter, superposing with incoming waves and forming standing waves with nodes and antinodes.

When two or more waves occupy the same space at the same time, they combine to form a single resultant wave.

The displacement of the resultant wave at any point is the sum of the displacements of the individual waves at that point. This is known as the principle of superposition.

Graphically, superposition can be represented by plotting the individual waves and then drawing a new line for their resultant. For each point on the wave, add the displacements of the individual waves to create the black line, which shows the resulting wave. This technique visually demonstrates the principle of superposition.

• Destructive interference: occurs when the peaks of one wave align with the troughs of another wave, resulting in cancellation or reduction in amplitude. Complete destructive interference happens when two equal but opposite waves cancel each other out.

• Constructive interference: occurs when the peaks (crests) of one wave align with the peaks of another wave, and their amplitudes add up to create a larger amplitude.

Interference occurs when two or more waves of the same type meet in space, and their resultant displacement is the sum of the displacements of each wave. This follows the principle of superposition, where the combined effect depends on the relative phase and amplitude of the waves involved.

The resultant wave may have a larger or smaller amplitude than the individual waves.

Types of interference:

• Constructive interference: occurs when two waves with the same frequency and amplitude are in phase (their peaks and troughs line up). The amplitude of the resultant wave is doubled.
• Destructive interference: occurs when two waves are in anti-phase (peaks of one wave align with troughs of the other). This causes the waves to cancel each other out, resulting in a wave with zero amplitude.

Coherent waves have the same frequency and a constant phase difference. This means that the peaks and troughs of the waves consistently match up at regular intervals. When waves are coherent, they produce stable, observable interference patterns because their phase relationship remains constant over time.

Non-coherent waves do not maintain a constant phase relationship. If two waves are not coherent, their phase relationship fluctuates. This means they do not consistently reinforce or cancel each other, which prevents consistent interference patterns from forming.

Practical examples:

• Coherent sources: a laser is an example of a coherent light source, where light waves have the same frequency and fixed phase difference, producing clear interference patterns.
• Incoherent sources: light from filament lamps is incoherent and produces waves with random phase relationships, which disrupt stable interference.

Path difference is the difference in the distances travelled by two waves from their sources to a particular point where they meet.

Path difference is crucial in determining whether the waves will interfere constructively or destructively when they meet. It is often expressed in terms of wavelength.

When two waves from different sources travel and meet at a point, the difference in the distances they travel (path difference) dictates the phase relationship between them.

• Constructive Interference occurs if the path difference is an integer multiple of the wavelength This is because each full wavelength difference corresponds to being “back in phase.”

• Destructive Interference happens when the path difference is a half-integer multiple of the wavelength: This leads to the peaks of one wave aligning with the troughs of the other, causing cancellation.

Phase difference is the difference in phase angle (measured in degrees or radians) between two waves that meet at the same point.

A phase difference reflects how ‘in step’ or ‘out of step’ two waves are. If two waves have no phase difference (0° or 0 radians), they are perfectly in phase, meaning their peaks and troughs align.

Path difference translates directly to phase difference because each full wavelength of path difference corresponds to a 360° (or radians) phase difference.

Examples

• A path difference of corresponds to a phase difference of leading to constructive interference.

• A path difference of corresponds to a phase difference of leading to destructive interference.

In this way, the phase difference in radians between two waves due to a path difference in metres is given by:

where is the wavelength in metres.

In a two-slit interference experiment, the waves from each slit travel slightly different distances to reach a point on the screen.

The wave from slit S2 has to travel further than the wave from slit S1 to reach the same point; the difference in these distances is the path difference.

This path difference determines whether the point on the screen will appear as a bright or dark fringe.

• If it is a bright fringe, this means constructive interference is occurring and the path difference is an even number of half wavelengths

• If it is a dark fringe, this means destructive interference is occurring and the path difference is an odd number of half wavelengths

Maxima and minima in an interference pattern:

• Maxima (bright fringes):

At positions where the path difference is an integer multiple of the wavelength, constructive interference occurs, creating a bright fringe.

These bright fringes are called maxima, and they are numbered by an order number , where is the central maximum, is the first order maximum on either side, and so on.

• Minima (dark fringes):

At positions where the path difference is a half-integer multiple of the wavelength, destructive interference occurs, creating a dark fringe.

These dark fringes, known as minima, are situated between the maxima on the interference pattern.

Young’s double-slit experiment is a classic demonstration of two-source interference using light waves. It shows how light, behaving as a wave, can produce an interference pattern.

1. A monochromatic light source shines through a single slit first, causing the light to diffract and spread out.
2. This light then passes through two narrow, closely spaced slits (A and B), which act as two coherent sources of light waves.
3. The light waves from each slit overlap and interfere as they travel to a screen, creating an interference pattern.

The interference of light waves from the two slits creates a pattern of bright and dark fringes on the screen.

• Bright fringes occur when the waves arrive in phase.
• Dark fringes occur where the waves arrive out of phase.

Because both slits are illuminated by the same initial light source, the light waves from slits A and B are coherent, having the same frequency and a constant phase difference. This coherence is essential for a stable, visible interference pattern.

In response to the limitations of Newton’s corpuscular theory of light, Christiaan Huygens developed the wave theory of light in the late 1600s.

According to Huygens, light behaves as a wave rather than as a stream of particles. So he introduced a concept now known as Huygens’ principle:

• Every point on a wavefront acts as a source of secondary wavelets that spread out in all directions at the same speed as the original wave.

• The new wavefront at any later time is found by drawing a surface tangential to these wavelets.

This principle enables the prediction of how waves propagate, including diffraction and refraction.

Advantages of the wave theory:

• Huygens’ wave model explained diffraction and interference, phenomena that Newton’s corpuscular theory could not account for.
• For example, Young’s double-slit experiment (1801) provided experimental confirmation of Huygens’ ideas, showing that light produces an interference pattern when passing through two slits; behaviour that is characteristic of waves.

The double-slit interference equation relates the distance between adjacent fringes (known as fringe width), the wavelength of the light, the distance between the slits, and the distance from the slits to the screen:

where:

• is the fringe width (the distance between two adjacent bright or dark fringes) in metres,
• is the wavelength of the light in metres,
• is the distance from the slits to the screen in metres, and
• is the distance between the two slits in metres.

A diagram is typically used to show the slit separation, the screen distance and the resulting interference fringes on the screen. This visual helps clarify the relationship between these variables in the experiment.

This equation is valid under the condition that slit separation, is much smaller than the screen distance, This condition ensures that the light waves from each slit reach the screen with a minimal change in angle.

Diffraction occurs when waves pass through a narrow gap (aperture) or around an obstacle, causing them to spread out as they pass through. Diffraction is a property of all types of waves, including sound, light, and water waves.

Waves are usually represented in diagrams as wavefronts, where each front shows a line of constant phase.

Diffraction is most pronounced when the gap size is comparable to or smaller than the wave’s wavelength. If the gap is much larger than the wavelength, the wave continues with minimal diffraction.

Wavelength, frequency, and wave speed remain unchanged during diffraction.

Diffraction of a wave leads to a diffraction pattern, which consists of maxima and minima intensity points.

For light waves, this corresponds to a series of dark and light fringes.

A diffraction pattern is visible when a laser is directed at a narrow slit where the slit width is larger than, but comparable to, the wavelength of the laser light. For laser light, a bright central fringe appears, surrounded by smaller fringes of decreasing brightness on either side.

If polychromatic white light is directed at a single slit (which is larger than visible light wavelengths), the central maximum will be white, and each fringe beyond it will show a spectrum of colours:

• Violet and blue light (shorter wavelengths) appear closest to the central maximum since they diffract the least.
• Red light (longer wavelengths) appears further out since it diffracts the most.

Dispersion refers to the process of splitting visible white light into its constituent colours, forming a spectrum. This can be achieved using either a glass prism or a diffraction grating.

A transmission diffraction grating is a tool often used in spectrometers for high-resolution separation of light by wavelength. It consists of a glass or plastic slide with many closely spaced, parallel slits or lines. When light passes through these slits, it diffracts and spreads out, allowing for detailed analysis of the light’s components.

For example, a continuous spectrum from a halogen light may be passed through a sample and then through a diffraction grating to produce an absorption spectrum. This absorption spectrum can then be used to identify the sample’s chemical composition.

Diffraction gratings are particularly useful for analysing light from stars. By separating the light into its individual wavelengths, scientists can determine the composition of stars based on their emission or absorption spectra.

To calculate the angular separation of each maxima, you can rearrange the diffraction grating equation to solve for

In this equation, is the angle measured from the centre (zero order) to the maxima.

Higher-order maxima (larger values of ) will occur at greater angles from the centre.

The angular separation between two maxima is simply the difference between their angles. For example, the separation between the first-order maxima, and second-order maxima is calculated as:

Refraction occurs when light passes from one medium to another with a different refractive index, such as from air to glass or from glass to water. This change in medium causes a change in the speed of light, which in turn alters its direction.

The direction of refracted light is measured relative to an imaginary line known as the normal, which is drawn perpendicular to the surface of the boundary where the light ray enters or exits.

The angle formed between the light ray and the normal before refraction is called the angle of incidence, while the angle after refraction is called the angle of refraction.

Refraction at a boundary depends on the refractive indices on either side:

• When light moves from a lower refractive index medium to a higher one, it slows down and bends towards the normal.
• When light moves from a higher refractive index medium to a lower one, it speeds up and bends away from the normal.
• If light travels directly along the normal line, no bending occurs because the light remains perpendicular to the boundary.

Only the speed and wavelength of light change during refraction. The frequency remains constant. This is a consequence of energy conservation.

When light refracts, its speed and wavelength change to match the properties of the new medium, but its frequency remains constant. This is a consequence of the conservation of energy.

A straw in a glass of water illustrates that frequency remains constant when it passes between two media. The appearance of the shape changes (the straw looks bent), but the colour of the straw remains the same in both air and water.

The frequency of light determines its colour (for example, red light has a wavelength of . Therefore, if the colour is the same, the frequency is the same.

Snell’s law is a mathematical relationship that describes how light refracts when it crosses the boundary between two media. It relates the angles of incidence and refraction to the refractive indices of the two media:

where:

• and are the refractive indices of the first and second media, respectively
• and are the angles of incidence and refraction, measured from the normal.

The first medium, where the light originates, is referred to as material 1, while the second medium, where the light enters, is referred to as material 2.

The angles in Snell’s law must always be measured relative to the normal.

Note that if a question gives an angle relative to the surface of the boundary, you will need to subtract it from to calculate the correct angle relative to the normal.

The refractive index of a transparent semi-circular block can be calculated by measuring the light passing through the block at various angles of incidence.

A semi-circular block is used because light passing through the curved surface always hits the boundary at normal incidence, so it does not refract. Therefore, if light is shone on the flat surface, the angle at which the light leaves the block can be measured to determine the angle of refraction from the flat surface.

Variables in this experiment:

• Independent variable: The angle of incidence . This is the angle at which the light ray strikes the flat surface of the semi-circular block.
• Dependent variable: The angle of refraction . This is the angle at which the light ray exits the block relative to the normal.

Experimental method:

1. Setting up the apparatus: Place the semi-circular block on a protractor such that its flat side lies along the line.
2. Defining the normal line: Direct the light ray from the lightbox or laser toward the flat edge of the block at . Using a ruler and pencil, draw a dotted line perpendicular to this edge to represent the normal.
3. Tracing the light rays: Direct the light ray into the block at various angles of incidence, starting from and increasing in increments up to . For each angle, trace the path of the ray entering the block and the ray exiting the block.
4. Measuring refraction: After tracing the light rays, remove the block to measure the angles of incidence and refraction relative to the normal using the protractor. Record these values in a table.
5. Repeating for accuracy: Perform multiple trials for each angle to calculate averages, reducing random errors.

From measurements of the angle of incidence and angle of refraction , the refractive index can be calculated using Snell’s law, which can be written as:

If we are dealing with a material surrounded by air, then represents the refractive index of air, which is approximately 1, and the equation simplifies to:

To accurately determine the refractive index, plot a graph of (Y axis) against (X axis). The gradient of this graph is equal to the refractive index of the block.

Several errors and uncertainties may affect the accuracy of the results.

Systematic errors:

• Ensure that the block of material that is being measured is positioned correctly. Misalignment can lead to consistent errors in the measurements of angles.

Random errors:

• Use a sharp pencil to draw lines accurately, reducing uncertainty in the measured angles.
• The light from the lightbox may appear slightly blurry or dispersed, making it difficult to trace the ray accurately. A concentrated laser beam produces sharper rays, reducing this uncertainty.

When light travels from a medium with a high refractive index to one with a lower refractive index, such as from glass to air, it refracts away from the normal.

As the angle of incidence increases, the angle of refraction also increases.

Eventually, as the angle of refraction increases, the refracted ray travels exactly along the boundary between the two media. This specific angle of incidence is known as the critical angle, denoted by

At the critical angle, the angle of refraction is

If the angle of incidence is greater than the critical angle, light no longer refracts into the second medium. Instead, it is completely reflected back into the denser medium. This phenomenon is called total internal reflection.

Total internal reflection has numerous practical applications: