Interference

1. Introduction

Interference of light is the phenomenon of the redistribution of light energy in a region where two (or more) light waves of the same frequency, travelling in the same direction, superpose. Where the waves arrive in phase the resultant intensity is a maximum (a bright fringe); where they arrive out of phase it is a minimum (a dark fringe).

• Interference was first demonstrated by Thomas Young (1801) and is direct evidence of the wave nature of light.
• It obeys the principle of superposition: the resultant displacement at a point equals the vector sum of the displacements due to each wave.
• Interference is fully consistent with the conservation of energy: no energy is created or destroyed, it is only redistributed from the dark fringes to the bright fringes.

2. Coherent Sources

Two sources of light are said to be coherent if they emit light waves of the same wavelength (and frequency) which maintain a constant phase difference with time (ideally zero, i.e. always in phase).

Key points:

• Two independent light sources can never be coherent, because light is emitted in random bursts, so their phase difference changes randomly and rapidly. No steady interference pattern is seen.
• Coherent sources are obtained in practice by deriving two waves from a single source (for example the two slits in Young's experiment, illuminated by the same monochromatic source). The two secondary sources then keep a fixed phase relationship.

Conditions for sustained (observable) interference:

1. The two sources must be coherent (constant phase difference).
2. They should emit light of the same wavelength/frequency (monochromatic).
3. The amplitudes should be equal or nearly equal for good contrast (fully dark minima).
4. The sources should be narrow and close together, and the screen reasonably far, so the fringes are wide enough to be seen.

3. Types of Interference

1. Constructive interference (bright fringe): when the two waves arrive at a point in phase, the resultant amplitude is the sum of the individual amplitudes and the intensity is maximum. This happens when the path difference is a whole number of wavelengths:

$\Delta x = n\lambda, \qquad n = 0, 1, 2, 3, \dots$

Since intensity $\propto (\text{amplitude})^2$, for two waves of amplitude $a$ the maximum amplitude is $2a$ and $I_{\max} \propto (2a)^2 = 4a^2$.

2. Destructive interference (dark fringe): when the two waves arrive out of phase (a crest meets a trough), the resultant amplitude is the difference of the amplitudes and the intensity is minimum. This happens when the path difference is an odd number of half-wavelengths:

$\Delta x = (2n-1)\frac{\lambda}{2}, \qquad n = 1, 2, 3, \dots$

For two equal waves the resultant amplitude is zero, giving $I_{\min} = 0$.

The phase difference $\phi$ and path difference $\Delta x$ are related by:

$\phi = \frac{2\pi}{\lambda}\,\Delta x$