Definition: Diffraction is the phenomenon of the bending of light around the corners of an obstacle or an aperture whose size is comparable to the wavelength of the light. Unlike interference, which involves the superposition of waves from two distinct coherent sources, diffraction arises from the interference of secondary wavelets originating from different parts of the same wavefront.
Fraunhofer Diffraction: The Far-Field Perspective
In optics, we categorize diffraction into two types: Fresnel and Fraunhofer. For competitive exams, Fraunhofer diffraction is the primary focus. This occurs when both the light source and the screen are effectively at an infinite distance from the aperture. In laboratory settings, we simulate this “infinity” by using convex lenses to render incoming light rays parallel and to focus the diffracted light onto a screen.
Because the rays are parallel, the phase difference between wavelets depends solely on the angle of diffraction. This simplifies the mathematical treatment significantly, allowing us to use plane wavefronts. When light passes through a slit, every point within the slit acts as a source of secondary wavelets, as per Huygens’ Principle. The resulting pattern on the screen is a distribution of intensity that varies based on the path difference between these wavelets.
Single-Slit Diffraction: The Intensity Pattern
When a monochromatic beam of light of wavelength λ passes through a single slit of width a, it produces a characteristic pattern on a screen. This pattern consists of a bright central maximum flanked by a series of alternating dark and weaker bright fringes. The central maximum is the brightest and widest part of the pattern, containing the majority of the light energy.
The condition for the formation of minima (dark fringes) is given by the formula:
a sin θ = nλ (where n = ±1, ±2, ±3, …)
Note that for n = 0, the equation gives sin θ = 0, which corresponds to the central maximum, not a minimum. The secondary maxima occur approximately midway between the minima, specifically at a sin θ ≈ (n + 0.5)λ. As the slit width a decreases, the central maximum spreads out, demonstrating the inverse relationship between the physical size of the aperture and the angular spread of the diffraction pattern.
The Rayleigh Criterion for Resolution
Diffraction imposes a fundamental limit on the ability of optical instruments, such as telescopes and microscopes, to distinguish between two closely spaced objects. This limit is defined by the Rayleigh Criterion. Two point sources are considered “just resolved” when the central maximum of one diffraction pattern falls exactly on the first minimum of the other.
For a circular aperture of diameter D, the angular resolution θ is given by:
- θ = 1.22 λ / D
If the angular separation between two objects is less than this value, the images will overlap to such an extent that they appear as a single blurred entity. This is why larger telescopes (with larger D) are superior; they provide a smaller θ, allowing for higher resolution and the ability to distinguish finer details in distant celestial bodies.
Important Facts and Formulas
| Concept | Formula / Condition |
|---|---|
| Single Slit Minima | a sin θ = nλ |
| Single Slit Maxima | a sin θ ≈ (n + 0.5)λ |
| Angular Resolution (Circular) | θ = 1.22 λ / D |
| Central Max Width | 2λf / a (for a lens of focal length f) |
Key Points to Remember
- Diffraction is most pronounced when the obstacle size is comparable to the wavelength (λ) of the light.
- The central maximum in single-slit diffraction is twice as wide as the secondary maxima.
- Increasing the slit width a narrows the diffraction pattern, making it approach the behavior of geometric optics.
- In the Rayleigh Criterion, the factor 1.22 arises specifically from the geometry of a circular aperture (Bessel function analysis).
- Diffraction explains why stars appear as points of light surrounded by diffraction rings in a telescope, rather than perfect geometric points.
- Intensity decreases rapidly as we move from the central maximum to higher-order secondary maxima.
Quick Revision Summary
- Diffraction results from the interference of secondary wavelets from a single wavefront.
- Fraunhofer diffraction assumes parallel incident and diffracted rays.
- Condition for dark fringes: a sin θ = nλ.
- The central maximum is the widest and most intense part of the pattern.
- Resolution is limited by the diffraction effect at the aperture.
- Rayleigh criterion: θ = 1.22 λ / D.
- Larger apertures lead to better resolution (smaller θ).
- Diffraction is the reason we cannot achieve infinite magnification in optical systems.
