 # Single Slit Diffraction

## What Is Diffraction of Light?

Diffraction is the bending of light around the sharp corner of an obstacle. When light is incident on a slit, with a size comparable to the wavelength of light, an alternating dark and bright pattern can be observed. This phenomenon is called the single slit diffraction. According to Huygens’ principle, when light is incident on the slit, secondary wavelets generate from each point. These wavelets start out in phase and propagate in all directions. Each wavelet travels a different distance to reach any point on the screen. Due to the path difference, they arrive with different phases and interfere constructively or destructively.

### Diffraction Due to Single Slit

When light is incident on the sharp edge of an obstacle, a faint illumination can be found within the geometrical shadow of the obstacle. This suggests that light bends around a sharp corner. The effect becomes significant when light passes through an aperture having a dimension comparable to the wavelength of light.

If light is incident on a slit having width comparable to the wavelength of light, an alternating dark and bright pattern can be seen if a screen is placed in front of the slit. This phenomenon is known as single slit diffraction.

### Young’s Single Slit Experiment

Thomas Young’s double slit experiment, performed in 1801, demonstrates the wave nature of light. In this experiment, monochromatic light is shone on two narrow slits. The waves, after passing through each slit, superimpose to give an alternate bright and dark distribution on a distant screen. All the bright fringes have the same intensity and width.

In a single slit experiment, monochromatic light is passed through one slit of finite width and a similar pattern is observed on the screen. Unlike the double slit diffraction pattern, the width and intensity in single slit diffraction pattern reduce as we move away from the central maximum.

### Explanation of The Phenomenon and Diffraction Formula

According to Huygens’ principle, when light is incident on the slit, secondary wavelets generate from each point. These wavelets start out in phase and propagate in all directions. Each wavelet travels a different distance to reach any point on the screen. Due to the path difference, they arrive with different phases and interfere constructively or destructively.

If a monochromatic light of wavelength $\lambda$ falls on a slit of width a, the intensity on a screen at a distance L from the slit can be expressed as a function of $\theta$. Here, $\theta$ is the angle made with the original direction of light. It is given by,

I($\theta$) =  $I_{o}$   $\frac{Sin^{2}\alpha}{\alpha^{2}}$

Here, $\alpha$  = $\frac{\pi}{\lambda}$ Sin $\theta$ and I0 is the intensity of the central bright fringe, located at $\theta$=0.

Diffraction Maxima and Minima: Bright fringes appear at angles,

$\theta$ → 0, $\theta$ → Sin-1 $\left (\pm \frac{3\lambda}{2} \right )$, $\theta$ → Sin-1 $\left (\pm \frac{5\lambda}{2} \right )$...

$\theta$ → 0 is the central maximum.

Dark fringes correspond to the condition,

a sin $\theta$ = m $\lambda$ with m =  $\pm$ 1,  $\pm$ 2, $\pm$ 3 ...

In a double slit arrangement, diffraction through single slits appears as an envelope over the interference pattern between the two slits.

### Fringe width

The angular distance between the two first order minima (on either side of the center) is called the angular width of central maximum, given by

2$\theta$ =  $\frac{2\lambda}{a}$

The linear width is as follows,

$\Delta$ = L . 2$\theta$ = $\frac{2L\lambda}{a}$

The width of the central maximum in diffraction formula is inversely proportional to the slit width. If the slit width decreases, the central maximum widens, and if the slit width increases, it narrows down. It can be inferred from this behavior that light bends more as the dimension of the aperture becomes smaller.

### Conditions for Diffraction

• The incident light should be monochromatic.

• The slit width should be comparable to the wavelength of incident light.

### Types of Diffraction

• Fresnel Diffraction: The light source and the screen both are at finite distances from the slit. The incident waves are not parallel.

• Fraunhofer Diffraction: The light source and the screen both are infinitely away from the slit such that the incident light rays are parallel.

### Solved Examples

1. Fraunhofer diffraction at a single slit is performed using a 700 nm light. If the first dark fringe appears at an angle 300, find the slit width.

Solution: Using the diffraction formula for a single slit of width a, the nth dark fringe occurs for,

a sin $\theta$ = n$\lambda$

At angle $\theta$ =300, the first dark fringe is located. Using n=1 and $\lambda$ = 700 nm=700 X 10-9m,

a sin 300=1 X 700 X 10-9m

a=14 X 10-7m

a=1400 nm

The slit width is 1400 nm.

1. Find the angular width of central maximum for Fraunhofer diffraction due to a single slit of width 0.1 m, if the frequency of incident light is 5 X 1014 Hz.

Solution: wavelength of the incident light is,

$\lambda$ =  $\frac{c}{v}$

Here, c=3 X 108m/s is the speed of light in vacuum and =5 X 1014Hz  is the frequency.

The angular width of the central maximum is,

2 $\theta$ = $\frac{2\lambda}{a}$

2 $\theta$ = $\frac{2c}{va}$

Using c=3 X 108m/s, =5 X 1014Hz and a=0.1 m,

2 $\theta$ = 1.2 X 10-4 rad

The angular width is 1.210-4 rad.

### Did you know?

• In the diffraction pattern of white light, the central maximum is white but the other maxima become colored with red being the farthest away.

• Diffraction patterns can be obtained for any wave. Subatomic particles like electrons also show similar patterns like light. This observation led to the concept of a particle’s wave nature and it is considered as one of the keystones for the advent of quantum mechanics.

• The interatomic distances of certain crystals are comparable with the wavelength of X-rays. Using X-ray diffraction patterns, the crystal structures of different materials are studied in condensed matter physics.