14.7 Diffraction

Diffraction Definition

Diffraction is a fundamental wave phenomenon where waves spread out and bend around obstacles or through openings. This behavior is what allows waves to travel around corners and propagate in directions beyond their initial path.

• Occurs with all types of waves including light, sound, water, and matter waves
• Enables sound to be heard around corners even when the source is not visible
• Explains how light can enter a room through small openings and illuminate areas not in direct line of sight
• Demonstrates the wave nature of light, contradicting the purely particle-based theories

Diffraction vs Opening Size

The relationship between wavelength and opening size determines how pronounced diffraction effects will be. This relationship is crucial for understanding when diffraction will significantly affect wave behavior.

• Maximum diffraction occurs when the opening size is approximately equal to the wavelength
• When opening width ≪ wavelength: waves spread out almost uniformly in all directions
• When opening width ≫ wavelength: minimal diffraction occurs, and waves travel mostly straight through
• Radio waves (λ ≈ meters) easily diffract around buildings and hills
• Visible light (λ ≈ 400-700 nm) shows minimal diffraction through doorways but significant diffraction through microscopic openings
• X-rays (λ ≈ 0.01-10 nm) require extremely narrow slits to produce observable diffraction

Interference Patterns

When waves diffract through openings, the resulting wavefronts can interfere with each other, creating distinctive patterns of intensity variations. These interference patterns are key to understanding and measuring wave properties.

• Diffracted wavefronts from different parts of the opening travel different distances to reach a point on the screen
• Constructive interference creates bright regions where waves arrive in phase (path difference = mλ)
• Destructive interference creates dark regions where waves arrive out of phase (path difference = (m+½)λ)
• The resulting pattern of bright and dark bands (fringes) is characteristic of wave behavior
• These patterns provide compelling evidence for the wave nature of light and matter
• The specific pattern depends on the shape of the opening and the wavelength of the wave

Single-Slit Diffraction Setup

A single-slit diffraction experiment demonstrates how waves behave when passing through a narrow opening. This classic setup reveals the fundamental principles of wave diffraction.

• Monochromatic light with wavelength λ passes through a narrow slit of width a
• Light travels distance L to a viewing screen
• Each point along the slit acts as a source of secondary wavelets (Huygens' Principle)
• These wavelets interfere to create a pattern of bright and dark fringes on the screen
• The central bright fringe is the widest and most intense
• Dark fringes occur at angles where path difference equals mλ
• For these dark fringes, the mathematical relationship is: $a\sin\theta = m\lambda$ (where m = 1, 2, 3...)
• For small angles, this approximates to: $a\left(\frac{y_{\min}}{L}\right) \approx m\lambda$
• Where y₍ₘᵢₙ₎ is the distance from the center to the mᵗʰ dark fringe

Diffraction Pattern Variations

The shape and arrangement of openings dramatically affect the resulting diffraction patterns. Different configurations produce characteristic patterns that can be analyzed to determine wave properties.

• Single circular aperture: produces a central bright spot surrounded by concentric rings (Airy disk)
• Rectangular slit: creates a pattern with bright regions extending perpendicular to the slit orientation
• Double slits: generate regularly spaced bright fringes modulated by a single-slit envelope
• Diffraction gratings (multiple slits): produce sharper, more intense bright spots at specific angles
• The intensity distribution varies with the specific geometry of the opening
• Complex aperture shapes create distinctive patterns that can be predicted using Fourier analysis

Visual Representations of Patterns

Visual representations help us understand and analyze diffraction patterns, providing insights into the wave properties and the diffracting structure.

• Intensity graphs show how brightness varies across the pattern
• The central maximum is typically much brighter than secondary maxima
• The width of the central bright fringe is inversely proportional to the slit width
• Narrower slits produce wider diffraction patterns
• The spacing between fringes is directly proportional to wavelength
• Measuring fringe spacing allows calculation of either wavelength or slit dimensions
• Comparing observed patterns with theoretical predictions confirms the wave model of light

Practice Problem 1: Single-Slit Diffraction

Solution

For a single-slit diffraction pattern, the position of the first dark fringe (m=1) is given by: $a\sin\theta = m\lambda$

For small angles, we can use the approximation: $\sin\theta \approx \tan\theta \approx \frac{y}{L}$

Where y is the distance from the center to the first dark fringe, and L is the distance to the screen.

Substituting into the equation: $a \cdot \frac{y}{L} = m\lambda$

For the first dark fringe (m=1): $a \cdot \frac{3.16 \times 10^{-3} \text{ m}}{2.0 \text{ m}} = 1 \cdot 632.8 \times 10^{-9} \text{ m}$

$a \cdot 1.58 \times 10^{-3} = 632.8 \times 10^{-9}$

$a = \frac{632.8 \times 10^{-9}}{1.58 \times 10^{-3}} = 4.0 \times 10^{-4} \text{ m} = 0.40 \text{ mm}$

Therefore, the width of the slit is 0.40 mm.

Practice Problem 2: Diffraction and Opening Size

Solution

Diffraction effects become minimal when the opening width is significantly larger than the wavelength of the wave. A common rule of thumb is that diffraction is negligible when the opening is at least 10 times the wavelength.

First, let's calculate the wavelength of the sound wave: $\lambda = \frac{v}{f}$

Where v is the speed of sound and f is the frequency.

$\lambda = \frac{340 \text{ m/s}}{680 \text{ Hz}} = 0.5 \text{ m}$

For diffraction effects to be minimal, the opening width should be at least 10 times the wavelength: $\text{Width}_{\text{min}} = 10 \times \lambda = 10 \times 0.5 \text{ m} = 5 \text{ m}$

Therefore, the opening should be at least 5 meters wide for diffraction effects to be minimal for this sound wave.