Interference of Coherent Light Waves

Young's Double-Slit Formula Used for Measurement of Wavelength

Aug 25, 2008

Young demonstrated a simple relationship between wavelength of light and the physical dimensions of the experimental setup, allowing for simple measurements of wavelength

If two light waves are in the same place at the same time, they combine with each other, either constructively or destructively.

In 1801 Thomas Young set up his famous double-slit experiment, showing that coherent light waves (waves with similar direction, amplitude, and phase) produce an interference effect. A pattern of light and dark bands, or fringes, may be seen when viewed on a screen. These bands of maximum and minimum light are depicted using colour code in Figure 1.

How Light Interference Occurs

For laser light interference to occur it must be passed through two very small slits placed extremely close to each other, as shown schematically in Fig 2. The pattern produced by light interference is very similar to that found with interference of sound or water waves: the dark bands, called nodes, correspond to cancellation, whilst the bright bands, called antinodes, correspond to constructive interference. Wave amplitudes combine according to the Principle of Superposition.

Conditions for Constructive or Destructive Interference

In Figure 2, coherent light shines onto two small apertures S1 and S2. If a crest (or trough) from S1 meets a crest (or trough) from S2, then reinforcement occurs. That is, crests are doubled (or troughs are doubled). The path difference PD from the source to the screen of each beam is equal to S2 - S1, which will always be zero or a whole number of wavelengths. Light bands or fringes (antinodes) will be seen at these points on the screen.

If a crest (or trough) from S1 meets a trough (or crest) from S2, cancellation occurs. The path difference PD is equal to S2 - S1, which will always be a half number of wavelengths. Dark bands or fringes (nodes) will be seen at these points on the screen.

Young's Formula

Figure 3 shows that the path difference is related to the angle of the first maximum subtended from the central maximum. For constructive interference at the first maximum, this equals to λ:

where W is the fringe width (the distance from the central maximum to the first maximum), and L is the distance from the source to the screen. For very small values of θ, sinθ and tanθ are approximately the same, so equating those leads to a simple formula for the measurement of wavelength in terms of the physical quantities of the double-slit apparatus:

Worked Example:

Light of an unknown wavelength emitted by a laser is directed through a pair of thin slits separated by 50 μm. The slits are 2m from a screen on which five bright fringes are seen over a width of 12.5 cm. Determine the wavelength of the light.

Solution: We know that W = 12.5 x 10‾² ÷ 5 = 2.5 x 10‾² m, L = 2 m, d = 50 μm. Using Young's formula and substituting in these values, one obtains λ = (2.5 x 10‾² x 5 x 10‾5)/2 = 6.3 x 10‾7 m. Another way to write this answer is 630 nm.

Summary

Thomas Young demonstrated that two light sources of coherent light interfere with each other depending on the phase difference between the wave trains. The wavelength of the light can be measured using a simple formula based only on the physical dimensions of the experimental apparatus.

The reader may be interested in more details of this topic, or learn about the nature of sound waves.

The copyright of the article Interference of Coherent Light Waves in Physics is owned by Harry P. Schlanger. Permission to republish Interference of Coherent Light Waves in print or online must be granted by the author in writing.