Diffraction Patterns and Interference (HSC SSCE Physics): Revision Notes

Diffraction Patterns and Interference

Introduction to light as a wave

In the early 19th century, experiments provided strong evidence that light behaves as a wave. The classical wave model, combined with electromagnetic field theory developed by Faraday and Maxwell, successfully explains many behaviours of light. Light is modelled as an electromagnetic wave that reflects, refracts, diffracts, and creates interference patterns, just like mechanical waves such as sound.

Young's double-slit experiment

Historical background

Thomas Young was a remarkable scientist born in 1773. He was extraordinarily gifted, teaching himself to read by age two and learning Latin by age six. At twenty-one, he was elected to the Royal Society. His interests spanned Egyptology, pure science, and medicine.

When Young read Newton's Opticks at seventeen, he was initially impressed with Newton's corpuscular theory (the idea that light consists of particles). However, by 1800, Young identified problems with this particle model:

• At certain boundaries, such as between water and air, some light was refracted while some was reflected. Particle behaviour could not easily explain this dual behaviour.
• The corpuscular theory failed to explain why different colours of light refract by different amounts.

The double-slit experiment explained

In the double-slit experiment, light passes through two narrow, closely spaced slits. The resulting interference pattern appears on a distant screen. This pattern arises because light waves from the two slits travel different distances to reach any given point on the screen.

The key concept is path difference - the difference in distance travelled by light from each slit to a point on the screen.

Understanding path difference

The path difference is given by:

$\delta = r_2 - r_1$

where $r_1$ and $r_2$ are the distances from each slit to a point on the screen.

When the screen is far from the slits, the two rays are approximately parallel. This allows us to express the path difference in terms of an angle $\theta$:

$\delta = d \sin\theta$

where $d$ is the separation between the slits.

Conditions for interference

Position of bright and dark spots

For small angles (when $L >> d$), we can relate the angle $\theta$ to the vertical position $y$ on the screen:

$\tan\theta \approx \sin\theta = \frac{y}{L}$

where $L$ is the distance from the slits to the screen.

The positions of bright spots are:

$y = \frac{Lm\lambda}{d}$

The positions of dark spots are:

$y = \frac{L\left(m + \frac{1}{2}\right)\lambda}{d}$

Determining wavelength

By measuring the distance between successive bright spots and knowing the slit separation, we can determine the wavelength:

$\lambda = \frac{dy}{L}$

Conversely, if we know the wavelength, we can calculate the slit separation. This makes Young's double-slit experiment a powerful tool for measuring wavelengths.

Single-slit diffraction

Fraunhofer diffraction

When light passes through a slit with finite width (not infinitesimally narrow), an interference pattern forms based on the width of the slit itself. This phenomenon is called Fraunhofer diffraction.

Conditions for single-slit minima

To understand single-slit diffraction, imagine dividing the slit into two halves. The first minimum (dark spot) occurs when each ray from the first half cancels with a corresponding ray from the second half.

Investigation 10.2: Single-slit diffraction

Aim: To observe the behaviour of light incident on a single slit.

Materials:

• Red laser pointer
• Laser mount
• Prepared slides with single slits of different widths
• Slide mount (boss head and clamp)
• Projection screen
• Metre ruler

Safety considerations:

What are the risks?	How can you manage these risks?
Temporary disturbance to vision from laser	Take strict precautions when using the laser. Do not point it at anybody. Do not look directly at the beam. Take particular care that all eyes are well out of the path of any reflections from the back of the slide.

Method:

1. Arrange the laser, a single-slit slide, and the screen as shown in the diagram.
2. Measure the distance from the slide to the screen.
3. Turn on the laser and observe the pattern formed on the screen.
4. Repeat for each of the other slides with different slit widths.

Results: Sketch each pattern observed. Note the relative brightness of the various spots.

Discussion: Consider how wave theory explains the observed patterns.

Combined double-slit and single-slit effects

In real double-slit experiments, the pattern combines contributions from both the slit width (Fraunhofer diffraction) and the separation between slits (double-slit interference). The overall pattern shows a diffraction envelope (determined by slit width) with interference substructure (determined by slit separation).

Number of interference fringes

We can calculate how many interference fringes appear within the central diffraction maximum.

For double-slit interference maxima: $d\sin\theta = m\lambda$

For single-slit minimum: $a\sin\theta = \lambda$ (when $m = 1$)

When the first single-slit minimum coincides with a double-slit maximum:

$m = \frac{d}{a}$

Investigation 10.3: Double-slit diffraction

This investigation has two parts.

Aim: To determine the distance between two slits on a microscope slide using double-slit diffraction.

Part 1: Determine the wavelength of the laser

Materials:

• Red laser pointer and laser mount
• Putty
• $1$ m thin, flexible wire
• Thin dowel rod
• Projection screen
• Metre ruler
• Wire cutters

Safety considerations:

What are the risks?	How can you manage these risks?
Temporary disturbance to vision from laser	Take strict precautions when using the laser. Do not point it at anybody. Do not look directly at the beam.
Cuts from razor blades	Carefully tape the blades together. When marking the slide, handle the blades via the clip.
Burns from candle flame	Waft the slide carefully through the yellow flame.

Method:

1. Place a small lump of putty on the bench about $2$ m from the screen.
2. Cut about $2$ cm off the thin flexible wire. Mount this vertically, protruding upwards from the putty.
3. Mount the laser so it points directly towards the mounted wire and the screen beyond.
4. Turn on the laser. This generates a diffraction pattern on the screen (a series of bright red dots).
5. Measure from the leftmost visible dot to the rightmost visible dot with the metre ruler.

1. Determine and record the average separation between dots (measure from centre to centre).
2. Wind the remaining wire onto the dowel rod, counting the number of windings carefully, until you have covered about $2$ cm of the dowel. Keep windings tight with no spacings or overlap.
3. Measure the distance from the first winding to the last along the dowel as precisely as possible.

1. Divide the measured distance by the number of windings to determine the thickness of the wire.

Results: Using the antinode separation distance and the thickness of the wire, calculate the wavelength of the laser using $\lambda = \frac{dy}{L}$.

Discussion: Red laser pointers typically produce wavelengths between 630 and 650 nm. Compare your calculated value to this expected range.

Part 2: Determine the distance between slits

Method:

1. Bind two razor blades together with sticky tape. Place them in a clip with a thin scrap of paper between them.
2. Light a candle (or Bunsen burner on yellow safety flame).
3. Carefully waft a microscope slide through the flame. The incomplete combustion produces carbon particles that coat the slide.
4. Using the bound razor blades, gently make a double slit on the carbon-covered slide.
5. Mount the slide with the slits vertical.
6. Shine the laser through the double slit onto the screen $2$ m away.
7. Determine and record the spacing between adjacent antinodes on the screen.

Results: Using the antinode separation distance and the calculated wavelength from Part 1, calculate the distance between the slits using $d = \frac{L\lambda}{y}$.

Analysis:

1. Determine the ratio of slit separation ($d$) to slit width ($a$).
2. Determine the slit width.
3. Calculate how many interference fringes appear within the first diffraction maximum using $N = 2m - 1$ where $m = \frac{d}{a}$.

Diffraction gratings

How diffraction gratings work

Diffraction gratings function similarly to double-slit experiments, but with many parallel slits very close together. When planar light passes through, diffraction spreads the light across numerous grating lines, each generating interference.

The path difference between adjacent lines remains:

$\delta = d\sin\theta$

The condition for maxima (bright spots) is:

$d\sin\theta = m\lambda, \quad m = 0, 1, 2, \ldots$

Intensity patterns

The location of maxima does not depend on the number of grating lines, $N$. However, the thickness of the maxima is inversely proportional to $N$. More grating lines produce thinner, sharper maxima.

Sources with mixed wavelengths (such as incandescent light) generate separate peaks for each wavelength, creating a spectrum.

Investigation 10.1: Interference patterns with incandescent light

Aim: To observe interference patterns from an incandescent light source using a diffraction grating.

Materials:

• Incandescent light source (desk lamp)
• Magnifying lens
• Diffraction grating (CD or DVD disc)
• Mount for disc
• Screen (wall, whiteboard, or large sheet of paper)

Safety considerations:

What are the risks?	How can you manage these risks?
Temporary disturbance to vision from magnified desk lamp	Do not look directly at the magnified light beam. Take care when reflecting the light beam off the diffraction grating that you do not stare directly at the undiffracted light.

Method:

1. Shine the incandescent light source through the magnifying glass so the beam focuses on the diffraction grating. Use the outer edge where the grooves are closest to parallel.
2. Observe the light diffracted by the grating.

Results: Sketch your observations, noting carefully any colour sequences.

Discussion: Explain the pattern of colours in terms of light as a wave, noting what happens to different wavelengths.

Investigation 10.4: Interference of monochromatic light

Aim: To observe interference of monochromatic light using a diffraction grating.

Materials:

• Red laser pointer and laser mount
• Diffraction grating (CD or DVD disc)
• Two-dimensional diffraction grating (nylon stocking)
• Diffraction grating mount
• Projection screen
• Metre ruler

Safety considerations:

What are the risks?	How can you manage these risks?
Temporary disturbance to vision from laser	Take strict precautions when using the laser. Do not point it at anybody. Do not look directly at the beam. Take particular care that when the laser is diffracted off the grating, all eyes are well out of the path of any reflections.

Method:

1. Set the diffraction grating parallel to the floor about $1$ m from the projection screen.
2. Shine the laser onto the CD diffraction grating so the beam is forward diffracted onto the projection screen. Use the outer edge where grooves are closest to parallel.
3. Observe the light pattern diffracted off the grating.
4. Measure the distances between spots on the screen.
5. Replace the CD grating with the two-dimensional grating and repeat.

Results: Sketch the patterns observed in both cases, indicating distances between selected spots.

Analysis: Using the wavelength determined in Investigation 10.3, calculate the spacing of the gratings in each case using $d = \frac{m\lambda}{\sin\theta}$.

Worked example: Calculating antinode separation

Remember!