Diffraction

SYLLABUS

Diffraction from a single slit, Diffraction pattern of image; Diffraction grating, Resolving power of optical instruments

What is This Chapter About?

Waves bend around obstacles. You can hear a conversation happening around the corner of a building even though you cannot see the person speaking — sound waves diffract around the corner and reach your ears. Light also diffracts, though the effect is much harder to notice with everyday objects because light’s wavelength is so small. But pass light through a narrow slit or around a sharp edge, and the bending and spreading become clearly visible.

Diffraction is the bending and spreading of waves when they encounter an obstacle or aperture whose size is comparable to the wavelength of the wave. It is one of the definitive signatures of wave behaviour — particles do not diffract — and it provides some of the most powerful tools in modern science, from X-ray crystallography that revealed the structure of DNA to diffraction gratings that analyse the composition of distant stars.

Chapter 12 of NEB Grade 12 Physics carries 3 teaching hours and covers single slit diffraction, diffraction gratings, and the resolving power of optical instruments — all topics with deep connections to experimental physics and optical technology.

Key Concepts You Will Learn

Diffraction at a Single Slit

When monochromatic light passes through a single narrow slit of width a, it spreads out and forms a diffraction pattern on a distant screen — a broad central bright maximum flanked by alternating dark and bright bands of decreasing intensity on both sides.

The dark fringes (minima) in the single slit diffraction pattern occur when:

[latex]a\sin\theta = n\lambda[/latex], where n = ±1, ±2, ±3, …

The central maximum occupies the region between the first minima on either side and is twice as wide as the secondary maxima. Its angular width is [latex]2\theta = \frac{2\lambda}{a}[/latex] — narrower slits produce wider diffraction patterns, and wider slits produce narrower patterns. This inverse relationship between aperture size and diffraction spread is a fundamental property of waves.

Understanding the Diffraction Pattern

The diffraction pattern can be understood using Huygens’ Principle from Chapter 10. Every point across the width of the slit acts as a source of secondary wavelets. At the centre of the screen, all wavelets travel equal distances and arrive in phase — constructive interference gives the bright central maximum. At angles where the path difference between wavelets from the top and bottom of the slit equals a whole wavelength, pairs of wavelets from different parts of the slit cancel in pairs — destructive interference gives the dark minima.

The intensity distribution across the single slit pattern follows:

[latex]I = I_0\left(\frac{\sin\beta}{\beta}\right)^2[/latex], where [latex]\beta = \frac{\pi a\sin\theta}{\lambda}[/latex]

This sinc-squared pattern has a dominant central peak and rapidly diminishing secondary peaks — the first secondary maximum has only about 4.7% of the central maximum intensity.

Diffraction Grating

A diffraction grating is an optical device consisting of a very large number of equally spaced parallel slits (typically thousands per millimetre). When light passes through a grating, interference between the many slits produces extremely sharp, bright maxima at specific angles — much sharper than the two-slit pattern of Young’s experiment.

The condition for bright maxima (principal maxima) from a diffraction grating is:

[latex]d\sin\theta_n = n\lambda[/latex]

where d is the grating spacing (distance between adjacent slits), [latex]\theta_n[/latex] is the angle of the nth order maximum, and n is the order number (0, ±1, ±2, …).

Rearranging: [latex]\sin\theta_n = \frac{n\lambda}{d}[/latex]. Since [latex]\sin\theta \leq 1[/latex], the maximum number of observable orders is limited by [latex]n_{max} = \frac{d}{\lambda}[/latex].

The grating equation shows that different wavelengths are diffracted at different angles — a diffraction grating disperses white light into its component colours, producing a spectrum. Unlike a prism, which produces a single spectrum through refraction, a grating produces multiple spectra (first order, second order, etc.) on both sides of the central maximum. Diffraction gratings are used in spectroscopes to measure wavelengths of light with extraordinary precision — they are the workhorses of modern spectroscopy.

Resolving Power of Optical Instruments

Two closely spaced objects can only be seen as separate if the optical instrument producing their images has sufficient resolving power. When light from a point source passes through a circular aperture (like a telescope or microscope lens), diffraction creates a circular bright disc surrounded by rings — the Airy disc. Two point sources can just be resolved when the central maximum of one Airy disc falls on the first minimum of the other — this is the Rayleigh criterion.

For a circular aperture of diameter D, the minimum angular separation that can be resolved is:

[latex]\theta_{min} = 1.22\frac{\lambda}{D}[/latex]

This result has profound implications. Larger telescope apertures resolve finer details because [latex]\theta_{min}[/latex] decreases with increasing D — this is why astronomical telescopes are made as large as possible. Using shorter wavelengths (like X-rays or electron waves) also improves resolution — this is the principle behind electron microscopes, which resolve features thousands of times finer than visible light microscopes.

Why This Chapter Matters for NEB Exams

Diffraction is examined thoroughly in NEB Grade 12 Physics. The single slit diffraction pattern — conditions for minima, width of central maximum, and the inverse relationship between slit width and diffraction spread — is a standard examination topic. The diffraction grating equation [latex]d\sin\theta = n\lambda[/latex] and its applications — finding wavelength, grating spacing, or angle — generate numerical problems that appear in virtually every board paper. The Rayleigh criterion for resolving power and its application to telescopes and microscopes is a frequently asked conceptual and numerical question.

Important Topics to Focus On

• Diffraction as bending of waves around obstacles — conditions for significant diffraction
• Single slit diffraction pattern — shape, central maximum, secondary maxima and minima
• Condition for minima in single slit diffraction: [latex]a\sin\theta = n\lambda[/latex]
• Angular width of central maximum: [latex]2\theta = \frac{2\lambda}{a}[/latex]
• Intensity distribution — sinc-squared pattern
• Effect of changing slit width on diffraction pattern
• Diffraction grating — construction and principle
• Grating equation: [latex]d\sin\theta_n = n\lambda[/latex]
• Maximum number of observable orders: [latex]n_{max} = \frac{d}{\lambda}[/latex]
• Dispersion of white light by a diffraction grating — comparison with prism
• Rayleigh criterion for resolution: [latex]\theta_{min} = 1.22\frac{\lambda}{D}[/latex]
• Resolving power of telescopes and microscopes
• Why larger apertures and shorter wavelengths improve resolution
• Numerical problems on single slit diffraction, grating equation, and resolving power

Connection to Other Chapters

Diffraction builds directly on Chapter 11 (Interference) — diffraction patterns are themselves interference patterns, arising from the superposition of wavelets from different parts of the aperture. Chapter 10 (Nature and Propagation of Light) provides Huygens’ Principle, which physically explains why secondary wavelets from the slit interfere to produce the diffraction pattern. Chapter 13 (Polarization) completes the wave optics section by demonstrating the transverse nature of light. The grating equation connects to Chapter 7 (Mechanical Waves) through the general wave equation [latex]v = f\lambda[/latex]. The resolving power of optical instruments is directly relevant to Chapter 17 of Grade 11 (Lenses) and the design of telescopes and microscopes. In modern physics, X-ray diffraction from crystal lattices — described by Bragg’s Law in Chapter 23 (Quantization of Energy) — is a direct extension of the diffraction grating concept to three-dimensional structures.