Diffraction

Diffraction is the phenomenon of the bending of waves around the edges of an obstacle or the spreading of waves as they pass through a narrow opening (aperture). Diffraction occurs when the wavelength of the wave is comparable to or larger than the size of the obstacle or aperture.

Diffraction is a wave phenomenon and provides further evidence for the wave nature of light (and other waves). It cannot be explained by the ray model of light, which predicts that light travels in straight lines.

According to Huygens' principle, every point on a wavefront acts as a source of secondary wavelets. When a wavefront is incident on an obstacle or aperture, the unobstructed portions of the wavefront continue to propagate. The secondary wavelets originating from the edges of the obstacle or aperture spread out into the region behind the obstacle or aperture, where the wave would not reach if it travelled only in straight lines. The superposition of these wavelets in the region behind the obstacle/aperture creates the diffraction pattern.

The Single Slit

A classic example of diffraction is the diffraction of light through a single narrow slit. When monochromatic light is incident on a narrow rectangular slit, instead of a simple bright band corresponding to the geometric shadow of the slit, a pattern of bright and dark fringes is observed on a screen placed behind the slit. This pattern is called a single-slit diffraction pattern.

$Diagram illustrating diffraction of plane waves by a single slit and the resulting pattern.$

(Image Placeholder: A screen with a single narrow slit of width 'a'. Plane wavefronts of monochromatic light are incident on the slit. Show the waves bending as they pass through the slit, spreading out. On a screen far behind the slit, show the diffraction pattern: a bright central maximum, flanked by less intense subsidiary maxima and dark minima.)

Explanation of the Single-Slit Pattern

Consider plane waves of monochromatic light of wavelength $\lambda$ incident normally on a slit of width $a$. According to Huygens' principle, every point in the slit acts as a source of secondary wavelets. These wavelets interfere on a distant screen.

Let's consider rays travelling from different points within the slit to a point P on the screen at an angle $\theta$ relative to the central axis.

Central Maximum: For $\theta = 0^\circ$, all rays from different points in the slit travel the same distance to the central point on the screen (O). The path difference is zero, so they interfere constructively, resulting in a bright central maximum.
Minima (Dark Fringes): Destructive interference occurs when the wavelets from different parts of the slit cancel each other out. This happens at angles $\theta$ such that the path difference between the wavelet from one edge of the slit and the wavelet from the opposite edge is an integer multiple of $\lambda$. More rigorously, consider dividing the slit into many small sources. The first minimum occurs when the path difference between the ray from the top edge and the ray from the bottom edge is $\lambda$. If the slit is divided into two halves, the rays from corresponding points in each half (separated by $a/2$) have a path difference of $\lambda/2$, leading to cancellation. This condition is satisfied when $a \sin\theta_n = n\lambda$, where $n = \pm 1, \pm 2, ...$. The first minima occur at $a\sin\theta_1 = \pm \lambda$, or $\sin\theta_1 = \pm \lambda/a$. For small angles, $\theta_1 \approx \pm \lambda/a$.
Subsidiary Maxima (Bright Fringes): Between the minima, the wavelets interfere constructively to produce bright fringes, but their intensity is much lower than the central maximum. These subsidiary maxima occur approximately halfway between the minima.

The condition for dark fringes (minima) in single-slit diffraction is:

$ a \sin\theta = n\lambda $

where $n = \pm 1, \pm 2, \pm 3, ...$ (excluding $n=0$, which corresponds to the central maximum).

The central bright maximum is much wider and more intense than the subsidiary maxima. Its width is approximately $2\lambda/a$ (the distance between the first minima on either side, for small angles). The widths of the subsidiary maxima are approximately $\lambda/a$.

The width of the central maximum is inversely proportional to the slit width $a$. A narrower slit produces a wider diffraction pattern. Also, the width of the pattern is proportional to the wavelength $\lambda$. Longer wavelengths (like red light) produce wider diffraction patterns than shorter wavelengths (like blue light).

Seeing The Single Slit Diffraction Pattern

To observe the diffraction pattern from a single slit clearly, certain conditions are helpful:

Monochromatic Light: Using light of a single wavelength (colour) simplifies the pattern. With white light, each colour diffracts by a different amount, resulting in overlapping coloured fringes, making the pattern less distinct.
Narrow Slit: The width of the slit ($a$) should be comparable to or smaller than the wavelength ($\lambda$) of the light ($a \approx \lambda$ or $a < \lambda$). If the slit is much wider than the wavelength ($a \gg \lambda$), the bending is negligible, and the geometric shadow is observed.
Distant Screen or Lens: The diffraction pattern is usually observed on a screen placed far away from the slit (Fraunhofer diffraction) or viewed through a lens placed near the slit (which focuses the diffracted rays).

You can qualitatively observe single-slit diffraction by looking through a very narrow slit (e.g., created by bringing two fingers close together) at a distant light source (like a street lamp at night). You might see fringes around the light source.

Resolving Power Of Optical Instruments

Diffraction limits the ability of optical instruments (like telescopes and microscopes) to distinguish between two closely spaced objects or details. This limitation is expressed by the instrument's resolving power.

Even if an optical system is perfect (free from aberrations), the image of a point object is not a perfect point due to diffraction at the aperture (the opening that limits the extent of the light beam). For a circular aperture (like the pupil of an eye or the objective lens of a telescope), the image of a point object is a diffraction pattern called the Airy pattern, consisting of a bright central disc (Airy disc) surrounded by fainter concentric rings.

When two point objects are very close together, their Airy patterns overlap. The two objects are said to be "resolved" (seen as separate) if the central maximum of one pattern falls at or beyond the first minimum of the other pattern. This is known as the Rayleigh criterion for resolution.

For a circular aperture of diameter $D$, the angle $\theta_{min}$ subtended by the first minimum of the diffraction pattern from the central maximum is given by:

$ \sin\theta_{min} \approx \theta_{min} = 1.22 \frac{\lambda}{D} $

According to the Rayleigh criterion, two point objects are just resolved if the angular separation between them is equal to $\theta_{min}$. The resolving power of the instrument is inversely proportional to this minimum angle of resolution. A smaller $\theta_{min}$ means higher resolving power.

Resolving Power of a Telescope: Ability to distinguish between two distant point sources (like stars). It depends on the diameter of the objective lens or mirror ($D$) and the wavelength of light ($\lambda$). Resolving power $\propto D/\lambda$. Larger diameter telescopes have higher resolving power. Using shorter wavelengths also improves resolving power (e.g., blue light telescopes, UV/X-ray telescopes).
Resolving Power of a Microscope: Ability to distinguish between two closely spaced points on a specimen. It depends on the wavelength of light ($\lambda$) and the numerical aperture (NA) of the objective lens, which is related to the angle of the cone of light entering the lens and the refractive index of the medium between the object and the lens. Resolving power $\propto NA/\lambda$. Higher NA objectives and shorter wavelengths improve resolving power. Electron microscopes use electron beams (much shorter wavelengths than light) to achieve much higher resolving power than optical microscopes.

Diffraction imposes a fundamental limit on the ability of any optical instrument to resolve fine details.

The Validity Of Ray Optics

Ray optics (geometrical optics) treats light as rays travelling in straight lines and undergoing reflection and refraction according to simple laws. This model is a simplification of the wave nature of light.

Ray optics is a good approximation and is valid when:

The wavelength of light is much smaller than the dimensions of the obstacles or apertures it interacts with ($\lambda \ll$ size of obstacle/aperture).
Diffraction effects are negligible.

When the size of the obstacle or aperture is comparable to or smaller than the wavelength of light, diffraction becomes significant, and the ray model breaks down. In such cases, the wave nature of light must be considered, and phenomena like interference and diffraction need to be explained using wave optics (physical optics).

For example, when light passes through large lenses or reflects from large mirrors, the bending and focusing can be accurately described using ray diagrams and lens/mirror formulas. However, if light passes through very narrow slits or around small obstacles, diffraction effects dominate, and the ray model is insufficient. The bending of light around a razor blade edge, for instance, is a diffraction effect.

The transition from ray optics to wave optics is gradual as the size of the object or aperture becomes closer to the wavelength. The study of diffraction helps us understand when the simpler ray model is applicable and when the more complex wave model is necessary.