Farunhofer Diffraction from a Single Slit and a Circular Aperture

Definition

Fraunhofer diffraction is the diffraction pattern observed when both the source and the screen are at effectively infinite distance from the diffracting aperture, or when a lens is used to convert the wavefronts into parallel rays. In this regime, the pattern depends on the angular distribution of light intensity rather than the near-field shape of the wavefront.

For a single slit of width $a$, Fraunhofer diffraction arises due to interference between light waves coming from different points across the slit. For a circular aperture of diameter $D$, the diffraction pattern results from the combined interference of secondary wavelets from the entire circular opening, producing a central bright spot known as the Airy disc.

Main Content

1. Fraunhofer Diffraction at a Single Slit

Basic idea and formation of the pattern

When monochromatic light passes through a narrow slit, every point across the slit acts as a source of secondary wavelets. In the far field, these waves travel nearly parallel to one another and interfere. Depending on the path difference between waves from different parts of the slit, the interference may be constructive or destructive. This gives rise to a pattern of one wide central maximum and alternate dark and bright fringes on both sides.

Conditions for minima and intensity distribution

The dark fringes occur when the path difference across the slit causes complete destructive interference. The condition for minima is
$$a \sin \theta = m\lambda \quad (m = \pm 1, \pm 2, \pm 3, \dots)$$ where $a$ is slit width, $\theta$ is the angle of observation, $\lambda$ is the wavelength, and $m$ is the order of minima.
The intensity distribution is given by
$$I = I_0 \left(\frac{\sin \beta}{\beta}\right)^2$$ where
$$\beta = \frac{\pi a \sin \theta}{\lambda}$$ This formula shows that the central maximum is the brightest and widest, while successive maxima decrease in intensity.

Key features of single-slit Fraunhofer diffraction

• The central maximum is twice as wide as each side maximum on either side.
• The angular width of the central maximum is approximately $2\lambda/a$ for small angles.
• Increasing slit width makes the diffraction pattern narrower, while decreasing slit width makes it broader.
• The pattern is evidence of the wave nature of light, since rays alone cannot explain the dark fringes.

2. Fraunhofer Diffraction at a Circular Aperture

Formation of the Airy pattern

When light passes through a circular opening such as a camera lens or telescope objective, diffraction occurs from every point in the aperture. The resulting pattern is not a set of parallel bright and dark bands but a bright central spot surrounded by concentric rings. This is known as the Airy pattern. The central bright region is called the Airy disc.

Condition for first minimum and resolving power

The first dark ring appears when the angle $\theta$ satisfies
$$\sin \theta = 1.22 \frac{\lambda}{D}$$ where $D$ is the diameter of the aperture. The factor 1.22 comes from the Bessel function solution for circular symmetry. The angular radius of the Airy disc is therefore
$$\theta \approx 1.22 \frac{\lambda}{D}$$ for small angles.
This result is crucial in optical resolution: two point objects are just resolved according to the Rayleigh criterion when the central maximum of one image coincides with the first minimum of the other.

Importance of circular aperture diffraction

• Determines the limit of resolution of optical devices.
• Larger apertures produce smaller Airy discs and better resolution.
• Shorter wavelengths also improve resolving power.
• It explains why even a perfect lens cannot form an infinitely sharp point image.

3. Comparison, Physical Interpretation, and Practical Meaning

Difference between single slit and circular aperture patterns

A single slit produces a line-like fringe pattern with alternating bright and dark bands, whereas a circular aperture gives a radially symmetric pattern of a bright central disc and rings. The geometry of the aperture directly shapes the diffraction pattern.

Role of wavelength and aperture size

In both cases, diffraction becomes more pronounced when the aperture is smaller or the wavelength is larger. Thus, red light diffracts more than blue light, and a narrow slit or small lens opening produces wider spreading of light. This relationship is expressed by the general idea that diffraction angle is proportional to $\lambda/\text{aperture size}$.

Wave theory explanation through interference

Fraunhofer diffraction is not caused by bending of light in a simple mechanical sense; rather, it is the result of superposition of wavelets from different parts of the aperture. The observed intensity pattern is a direct consequence of constructive and destructive interference in the far field.

Working / Process

1. Illuminate the aperture with monochromatic light

A coherent and single-wavelength source, such as a laser, is used so that stable interference fringes are formed. The slit or circular opening should be placed so that the incoming wavefront is effectively plane.

2. Use far-field observation or a lens arrangement

In practice, a converging lens is often placed after the aperture. The diffraction pattern is then observed in the focal plane of the lens, which corresponds to the Fraunhofer region. This makes the angular distribution visible on a screen.

3. Observe and analyze the intensity pattern

For a single slit, measure the positions of minima and maxima to determine slit width or wavelength. For a circular aperture, measure the size of the central bright spot and rings to estimate aperture diameter and resolving power. The observed pattern is then compared with the theoretical conditions:

• Single slit: $a\sin\theta = m\lambda$
• Circular aperture: $\sin\theta = 1.22\lambda/D$

Advantages / Applications

Explains the resolution limits of optical instruments

Fraunhofer diffraction is essential in understanding why telescopes, microscopes, and cameras cannot produce perfectly sharp images beyond a certain limit.

Used in measurement of wavelength and aperture size

By analyzing diffraction fringes, one can determine the wavelength of light, slit width, or aperture diameter with good accuracy in laboratory experiments.

Important in engineering, astronomy, and optical design

It is used in designing lenses, apertures, spectrometers, and imaging systems, and in astronomy to understand the appearance of stars and the resolving power of telescopes.

Summary

• Fraunhofer diffraction is the far-field diffraction pattern produced by a slit or aperture when wavefronts interfere at large distances or in the focal plane of a lens.
• A single slit gives a central bright maximum with weaker side maxima and minima at $a\sin\theta = m\lambda$.
• A circular aperture produces an Airy disc with rings, and its first minimum occurs at $\sin\theta = 1.22\lambda/D$.
• The topic explains diffraction patterns, wave interference, and the resolving power of optical instruments.