Question

3. State salient features of single slit Fraunhofer diffraction pattern. The slit is vertical and
illuminated by a point source. Also, obtain an expression for intensity distribution and plot it.

Answer 1

Answer on Question #50695, Physics, Optics

3. State salient features of single slit Fraunhofer diffraction pattern. The slit is vertical and illuminated by a point source. Also, obtain an expression for intensity distribution and plot it.

Solution:

The Fraunhofer diffraction equation is used to model the diffraction of waves when the diffraction pattern is viewed at a long distance from the diffracting object.

The diffraction at a single slit of width $d$ is shown in Figure 1. Diffraction occurs in all directions to the right of the slit.

Fig.1. Graph and image of single-slit diffraction

The pattern consists of a central bright fringe (band) flanked by much weaker maxima alternating with dark fringes.

The general condition for a minimum for a single slit is:

$$m \lambda = a \sin \theta$$

where $m = 1,2,3,4$ and so on

- $a$ is the width of the slit,

- $\theta$ is the angle at which the minimum intensity occurs, and

- $\lambda$ is the wavelength of the light

These two diagrams show the effect of a change of wavelength on the single slit diffraction pattern. The pattern for red light is broader than that for blue because of the longer wavelength of red light.

One of the characteristics of single slit diffraction is that a narrower slit will give a wider diffraction pattern as illustrated below, which seems somewhat counter-intuitive.

Intensity of single-slit diffraction patterns

We can use phasors to determine the light intensity distribution for a single-slit diffraction pattern. Imagine a slit divided into a large number of small zones, each of width $\Delta y$ as shown in Figure. Each zone acts as a source of coherent radiation, and each contributes an incremental electric field of magnitude $\Delta E$ at some point $P$ on the screen. We obtain the total electric field magnitude $E$ at point $P$ by summing the contributions from all the zones. The light intensity at point $P$ is proportional to the square of the magnitude of the electric field.

The incremental electric field magnitudes between adjacent zones are out of phase with one another by an amount $\Delta \beta$ , where the phase difference $\Delta \beta$ is related to the path difference $\Delta y$ sin $\theta$ between adjacent zones by the expression

$$\Delta \beta = \frac {2 \pi}{\lambda} \Delta y \sin \theta$$

The total phase difference $\beta$ between waves from the top and bottom portions of the slit is

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