Wave-Like Properties of Light (VCE SSCE Physics): Revision Notes

Wave-Like Properties of Light

Introduction to electromagnetic waves

Light is a type of electromagnetic wave - a transverse wave consisting of perpendicular oscillating electric and magnetic fields. This understanding emerged from James Clerk Maxwell's pioneering work in 1826, when he demonstrated mathematically that the laws of electricity and magnetism predicted the existence of electromagnetic waves.

Key properties:

• Electric and magnetic fields oscillate at right angles to each other
• Both fields are perpendicular to the direction of wave travel
• Speed in vacuum: $3.00 \times 10^8 \, \text{m} \cdot \text{s}^{-1}$
• Wavelengths range from $10^{-18} \, \text{m}$ to more than $10^4 \, \text{km}$

The wave equation

The relationship between speed, frequency, and wavelength for electromagnetic waves is given by:

$c = f\lambda$

Where:

• $c$ = speed of light in vacuum ($3.0 \times 10^8 \, \text{m} \cdot \text{s}^{-1}$)
• $f$ = frequency (Hz)
• $\lambda$ = wavelength (m)

When electromagnetic waves travel through a medium (such as glass or water), they move more slowly than in a vacuum. The relationship is:

$v = \frac{c}{n}$

Where:

• $v$ = speed of light in the medium (m·s⁻¹)
• $n$ = refractive index of the medium
• $c$ = speed of light in vacuum

Applications of lasers

Lasers are a key technology demonstrating light's properties with numerous applications across multiple fields:

• Medicine: retinal repair, laser scalpels (incisions as narrow as $5 \times 10^{-4}$ m), dental procedures
• Measurement: distance measurement, surveying, Earth-Moon distance monitoring (accurate to 15 cm)
• Industry: cutting and welding in engineering and garment construction
• Communications: optical fibres using infrared laser light
• Technology: DVD readers, barcode scanners, 3D scanning, LIDAR mapping
• Research: LIGO detectors for gravitational waves
• Agriculture: autonomous weeding systems

Standing waves

A standing wave (also called a stationary wave) is a wave pattern that oscillates in time but whose amplitude profile does not move in space. Standing waves form when two equal waves travelling in opposite directions meet and overlap.

Formation of standing waves

Standing waves can be produced in a stretched string by fixing one end and vibrating the other end. The reflected wave overlaps with the original wave, and when the frequency meets certain conditions, a standing wave pattern emerges.

Formation through superposition

Standing waves result from the principle of superposition - when two or more waves overlap, their displacements add together. The diagrams show how forward and reflected waves combine at different times:

Position a: The incident and reflected waves are in phase, producing maximum displacement (constructive interference).

Position b: The waves are out of phase, cancelling each other to produce no displacement (destructive interference).

Position c: The waves again reinforce, producing maximum displacement in the opposite direction.

Conditions for standing waves with nodes at both ends

This principle applies to both transverse waves (like waves on a string) and longitudinal waves (like sound waves in a pipe open at both ends).

Harmonic series

When standing waves form in a stretched string, they create a series of patterns called the harmonic series. Each harmonic has its own frequency:

The first harmonic (or fundamental) is the simplest pattern, where the string length equals half a wavelength: $l = \frac{\lambda}{2}$

The frequency of the fundamental is: $f_1 = \frac{v}{\lambda} = \frac{v}{2l}$

For the second harmonic, the string length equals one full wavelength: $l = \lambda$ $f_2 = \frac{v}{l}$

Standing light waves

Standing waves can also form with light. Inside a laser cavity, light reflects between mirrors and can establish a standing wave pattern:

Diffraction

Diffraction is the spreading of a wave when it passes through a narrow opening or around an obstacle. While easily observed with sound and water waves, diffraction also occurs with light, though it can be harder to see.

When a laser beam passes through a narrow slit, it spreads out. The amount of spreading depends on:

• The wavelength of light ($\lambda$)
• The width of the slit ($w$)
• The distance to the screen ($L$)

The width of the diffracted beam on the screen is proportional to $\frac{\lambda L}{w}$.

Effect of slit width on diffraction

As a rough guide, when $\frac{\lambda}{w} = 0.01$, the beam spreads at about 1°.

Intensity distribution

Diffraction patterns can be represented as graphs showing light intensity versus position on the screen:

Diffraction around obstacles

Diffraction also occurs when waves encounter obstacles:

The amount of diffraction around an obstacle depends on the ratio $\frac{\lambda}{w}$ (where $w$ is the obstacle width):

• When $\frac{\lambda}{w} < 1$ but not $\ll 1$: clear diffraction occurs
• When $\frac{\lambda}{w} \approx 1$: significant diffraction occurs
• When $\frac{\lambda}{w} \ll 1$: very little diffraction occurs

Applications to resolving close objects

Diffraction limits our ability to distinguish between two close light sources. Consider viewing car headlights at different distances:

Distant headlights: The diffracted images overlap completely, appearing as a single light source.

Intermediate distance: The images begin to separate but still overlap partially.

Close headlights: The images are clearly separated and distinguishable as two distinct sources.

Diffraction and astronomical telescopes

Diffraction affects all telescopes, including space telescopes like the Hubble and James Webb telescopes:

The ability to separate close astronomical objects depends on the $\frac{\lambda}{w}$ ratio, where $w$ is the telescope aperture (diameter). To reduce diffraction effects:

Increase aperture size: Larger telescopes have smaller diffraction. Many terrestrial reflecting telescopes have diameters greater than 10 m.

Use shorter wavelengths: Observing in ultraviolet rather than visible light reduces diffraction.

Alpha Crucis example: The bottom star of the Southern Cross, Alpha Crucis, is actually a double star. A small telescope can resolve the two stars, but the naked eye (with pupil diameter about 5 mm) sees only a single star because the diffracted images overlap.

Huygens' principle

More than a century before the wave model was established, scientist Christiaan Huygens proposed a principle for light propagation. He suggested that every point on a light wavefront acts as a source of secondary wavelets (wave-like phenomena that begin with zero amplitude). The envelope of all these wavelets forms the new wavefront.

For a plane wavefront:

• Points on the original wavefront each generate secondary wavelets
• The tangent line to these wavelets forms the new wavefront
• The wavefront propagates forward

The Poisson spot

A remarkable demonstration of diffraction is the Poisson spot - a bright spot appearing at the centre of the shadow cast by a circular object:

Explanation: Light diffracts around the edges of the circular object. Since all points on the edge are equidistant from the centre of the shadow, the diffracted wavefronts arrive at the centre simultaneously and in phase. This creates constructive interference, producing a bright spot.

Young's double slit experiment

At the start of the 19th century, debate about the nature of light was dominated by Isaac Newton's particle theory. Newton believed light consisted of coloured particles called "corpuscles" that travelled in straight lines at great speed. This theory could explain reflection, refraction, and dispersion of white light into colours.

However, Thomas Young's famous double slit experiment established light's wave-like nature and turned the tide of scientific opinion.

Original experiment

In Young's original experiment, he allowed a narrow beam of sunlight to be cut in two by a thin card, creating two light sources:

When the light from these two sources overlapped (due to diffraction), a series of dark and bright bands appeared, with the bright fringes edged with colours.

Modern setup

Modern versions typically use a monochromatic laser instead of sunlight, producing a clearer pattern:

The setup consists of:

• A laser beam with wavelength $\lambda$
• Two parallel slits separated by distance $d$ (approximately 0.5 mm)
• A screen at distance $L$ (typically several metres away)
• An interference pattern of alternating bright and dark bands

Understanding the interference pattern

At point C (centre):

• Bright band appears
• Light from slits $S_1$ and $S_2$ travels exactly the same distance
• Waves arrive in phase (peaks and troughs matching)
• Constructive interference occurs - the intensities add together

At point X (third bright band):

• The distance $S_2X$ exceeds distance $S_1X$ by exactly three wavelengths: $S_2X - S_1X = 3\lambda$
• Waves still arrive in phase
• Constructive interference produces a bright band

At point Y (second dark band):

• The path difference is: $S_1Y - S_2Y = \frac{3\lambda}{2}$
• Waves arrive out of phase
• Destructive interference occurs - the waves cancel each other

Conditions for interference

Fringe spacing formula

The distance between adjacent bright (or dark) bands is given by:

$\Delta x = \frac{L\lambda}{d}$

Where:

• $\Delta x$ = distance between bright bands (or dark bands) (m)
• $L$ = distance between slits and screen (m)
• $d$ = separation between slits (m)
• $\lambda$ = wavelength of light (m)

Coherence

For clear interference patterns to appear, the two light sources must be coherent - they must be monochromatic (single wavelength) with a fixed or zero phase difference between them.

Young's original experiment used sunlight passing through a single narrow slit before reaching the double slits. This introduced sufficient coherence to produce visible interference bands. Modern demonstrations typically use lasers, which are fully coherent and produce clearer, brighter patterns.

Evidence for wave-like nature of light

Young's double slit experiment provided compelling evidence for light's wave-like nature by demonstrating that light behaves identically to other established wave phenomena.

Comparison with other waves

The same interference patterns observed with light can be demonstrated with:

Sound waves:

Two loudspeakers producing the same frequency note in phase create regions of:

• Maximum sound intensity (antinodes) where path differences equal whole wavelengths
• Minimum sound intensity (nodes) where path differences equal odd multiples of half wavelengths

Water waves:

Two dippers in a ripple tank producing circular waves in phase create:

• A central antinode where path difference is zero (constructive interference)
• Alternating nodes and antinodes where path differences vary by half wavelengths

Additional wave-like evidence

Beyond Young's experiment, other observations support light's wave-like nature:

• Diffraction: Light spreads when passing through narrow openings or around obstacles
• Light beams pass through each other without interaction
• Reflection and refraction: Light reflects and refracts according to wave principles
• Polarisation: A property unique to transverse waves
• Dispersion: White light separates into component colours (spectrum)
• Light travels in straight lines (consistent with wave propagation)

The wave model successfully explains all these phenomena, whereas the particle model struggled to account for diffraction, polarisation, and the results of Young's experiment.

Remember!