Question

Can light pass through a single slit narrower than its wavelength? If not, why not? If so, describe the distribution of the light beyond the slit.

Short answer

If yes, how is the distribution of light beyond the slit? Answer: Yes, light can pass through a single slit narrower than its wavelength, as it undergoes diffraction. The distribution of light beyond the slit consists of a central bright spot followed by alternating bright and dark regions. The intensity distribution of the diffracted light can be described using the formula: \(I(\theta)= I_0 \left[\frac{\sin \left(\frac{\pi a \sin \theta}{\lambda}\right)}{\frac{\pi a \sin \theta}{\lambda}}\right]^2\) where \(I(\theta)\) is the intensity of light at angle \(\theta\), \(I_0\) is the maximum intensity of the central bright spot, and \(a\) is the width of the slit.

Step-by-step solution

Plane waves incident on the single slit

Before discussing the diffraction phenomenon, we begin with plane waves incident on the single slit. Plane waves carry uniform energy in a particular direction and have no constraints on their amplitude. The slit acts as a barrier that either blocks or allows the light to pass through.

Huygens' Principle and Diffraction

Huygens' Principle states that each point on a wavefront can be considered a source of secondary wavelets that spread out in all directions. Diffraction is the phenomenon in which light waves bend around obstacles, such as the edges of the slit, giving rise to secondary waves. When the slit width (denoted by "a") is larger or comparable to the wavelength (denoted by "λ") of light, these secondary waves interfere constructively and destructively, creating a pattern of varying intensity.

Diffraction through a single slit

When light passes through a single slit narrower than its wavelength (a < λ), diffraction becomes more pronounced. The secondary waves spread out more widely and interfere constructively and destructively beyond the slit. The result is a distribution of light known as a diffraction pattern, characterized by a central bright spot or maximum surrounded by alternating bright and dark regions.

Distribution of light beyond the narrow slit

The distribution of light beyond a narrow single slit can be described mathematically with the following formula: \[ I(\theta)= I_0 \left[\frac{\sin \left(\frac{\pi a \sin \theta}{\lambda}\right)}{\frac{\pi a \sin \theta}{\lambda}}\right]^2 \] where \(I(\theta)\) is the intensity of light at angle \(\theta\), \(I_0\) is the maximum intensity of the central bright spot, and \(a\) is the width of the slit. This formula gives the intensity distribution of light as a function of the angle \(\theta\). In conclusion, light can pass through a single slit narrower than its wavelength, and it undergoes diffraction giving rise to a pattern characterized by a central bright spot followed by alternating bright and dark regions. The intensity distribution of the diffracted light can be described by the formula presented above.