## Variation of intensity with angular position

To find an expression for the intensity in terms of angular position, we need to calculate the total amplitude of Huygens' wavelets arriving at different points on the screen. The wavelets are assumed to leave the slit in phase, so phase differences at the screen are related solely to path differences.

A path difference of A/2 corresponds to a phase difference of n.

In general, phase difference = — x (path difference)

For light travelling at an angle d from two adjacent sources separated by Ax, the path difference = Dx sine.

Phase difference between adjacent wavelets A j = Ax sin0.

We use a vector technique called the addition of phasors, in which the slit is treated as a very large number of identical and equally spaced sources of Huygens' wavelets. Each wavelet is represented by a phasor, whose length represents the amplitude of the electric field. For identical sources, all phasors have the same length. The phase difference between wavelets is represented by the angle between the corresponding phasors.

To sum over all the wavelets arriving at any point on the screen, we arrange the phasors from adjacent sources head to tail in vector fashion, starting at the top of the slit. The total amplitude is given by the line joining the two ends of the 'phasor chain', as illustrated in Figure 8.25.

At the centre of the pattern 6 = 0, and all wavelets are in phase.

The amplitude has its maximum value E0

Figure 8.25 Phasor diagrams for five wavelets.

For very large numbers of sources, the individual amplitudes are so small that the phasor chain forms a smooth circular arc. We can express E in terms of E0 using Figure 8.26.

Intensity ^ (amplitude)

sin a a sin2 a

Figure 8.27 shows how the intensity varies across the screen as a function of the angle and includes the angular positions of minima.

The angular separation of the minima on either side of the central peak is often referred to as the angular width of the main maximum.

Angular width of the main maximum = —

a intensity I

intensity I

-L position 0

3A a

3A a

Figure 8.27 Graph of intensity as a function of angular position.

3A a

3A a

-L position 0

Figure 8.27 Graph of intensity as a function of angular position.

0 0