4.4.6 Interference & Path Difference

• Interference occurs whenever two or more waves combine to produce a resultant wave with a new resultant displacement
• The waves combine according to the principle of superposition
  • Constructive interference happens when the resultant wave has a larger displacement than any of the individual displacements
  • Destructive interference happens when the positive displacement of one wave and the negative displacement of another wave exactly cancel out giving a resultant displacement of zero 

Coherence

• Interference is only observable if produced by a coherent source
• Waves are said to be coherent if they have:
  • A constant phase difference
  • The same frequency

Coherent waves (on the left) and non-coherent waves (on the right). The abrupt change in phase creates an inconsistent phase difference

• A coherent beam of light contains light waves that are monochromatic and have a constant phase difference
  • Monochromatic light consists of light waves of a single frequency 

• Laser light is an example of a coherent light source
• Filament lamps produce incoherent light waves

Double-Slit Interference of Light

• When a coherent beam of light is incident on two narrow slits very close together, diffraction occurs at each slit (i.e. the waves spread out)
• As the diffracted waves cross, they interfere with each other
• If a screen is placed some distance away from the slits, a pattern of equally spaced bright and dark fringes is observed on the screen
  • The bright fringes form where the waves interfere constructively (i.e. a crest meets a crest or a trough meets a trough)
  • The dark fringes form where the waves interfere destructively (i.e. a crest meets a trough)

Coherent light waves interfere after passing through two narrow slits. Alternating bright and dark fringes are observed on the screen

• The type of interference occurring at a given point (i.e. constructive or destructive) depends on the path difference of the overlapping waves
• Path difference is defined as:

The difference in distance travelled by two waves from their sources to the point where they meet

• Path difference is generally expressed in multiples of wavelength

At point P₂ the waves have a path difference of a whole number of wavelengths resulting in constructive interference. At point P₁ the waves have a path difference of an odd number of half wavelengths resulting in destructive interference 

• In the diagram above, the number of wavelengths between:
  • S₁ ➜ P₁ = 6λ
  • S₂ ➜ P₁ = 6.5λ
  • S₁ ➜ P₂ = 7λ
  • S₂ ➜ P₂ = 6λ

• The path difference is:
  • (6.5λ – 6λ) = λ / 2 at point P₁
  • (7λ – 6λ) = λ at point P₂

• Hence:
  • Destructive interference occurs at point P₁
  • Constructive interference occurs at point P₂

Conditions for Constructive and Destructive Interference

• In general, for waves emitted by two coherent sources very close together:
  • The condition for constructive interference is:

path difference = nλ

• • The condition for destructive interference is:

path difference = (n + ½)λ

• • Where:
    • λ = wavelength of the waves in metres (m)
    • n = 0, 1, 2, 3... (any other integer)

• The same conditions apply to waves emitted by a single coherent source and diffracted by two narrow slits very close together

Path Difference and Wavefronts

At point P the waves have a path difference of a whole number of wavelengths resulting in constructive interference

• Another way to represent waves spreading out from two sources is shown in the diagram above
• At point P, the number of crests from:
  • Source S₁ = 4λ
  • Source S₂ = 6λ

• The path difference at P is (6λ – 4λ) = 2λ
• This is a whole number of wavelengths (n = 2), hence constructive interference occurs at point P

Step 1: Count the number of wavelengths between each source and the desired point 

• • E.g. Number of wavelengths between:
    • S₁ ➜ A = 5λ
    • S₂ ➜ A = 6.5λ

Step 2: Determine the path difference by subtracting the distances of the point from the two sources 

• • E.g. Path difference at A = (6.5λ – 5λ) = 1.5λ

Step 3: Compare the path difference calculated in Step 2 with the condition for constructive or destructive interference and give the value of n 

• • E.g. Path difference at A = 1.5λ = (n + ½)λ ➜ n = 1

Step 4: Decide whether the point is a location of constructive or destructive interference 

• • E.g. A is a location of destructive interference

• Point A:
  • Path difference = (6.5λ – 5λ) = 1.5λ
  • n = 1
  • Destructive interference

• Point B:
  • Path difference = (5λ – 4λ) = λ
  • n = 1
  • Constructive interference

• Point C:
  • Path difference = (2λ – 2λ) = 0
  • n = 0
  • Constructive interference

• Point D:
  • Path difference = (5λ – 4.5λ) = 0.5λ
  • n = 0
  • Destructive interference

• Point E:
  • Path difference = (4λ – 3λ) = λ
  • n = 1
  • Constructive interference