14.8 Double-Slit Interference and Diffraction Gratings

Double-slit interference and diffraction gratings showcase light's wave nature. These phenomena produce intricate patterns of bright and dark bands due to constructive and destructive interference of light waves from multiple sources.

Understanding these concepts helps explain various optical effects and has practical applications. Diffraction gratings, with many evenly spaced slits, create complex interference patterns used in spectrometers and lasers for wavelength analysis and selection.

Wave Behavior and Diffraction Patterns

Monochromatic Light and Double Slits

When monochromatic light (light of a single wavelength) passes through two narrow slits, it creates a distinctive pattern on a distant screen that demonstrates both diffraction and interference effects. 🌊

• Wave interference occurs when waves from the two slits overlap and combine:

  • Constructive interference (bright bands) happens when waves arrive in phase
  • Destructive interference (dark bands) occurs when waves arrive out of phase
  • The interference pattern depends on the path length difference (ΔD) between waves from each slit

• The path length difference relates to the slit separation (d) and angle (θ) by: $\Delta D = d \sin \theta$

• For small angles (θ < 10°), the position of the mth bright fringe (maximum) can be found using: $y_{max} = \frac{m\lambda L}{d}$

  Where:

  • y₍ₘₐₓ₎ is the distance from the central maximum
  • λ is the wavelength
  • L is the distance to the screen
  • d is the slit separation
  • m is the order number (0, 1, 2, etc.)

• The complete pattern combines interference (creating evenly spaced maxima) with single-slit diffraction (creating an intensity envelope that modulates the brightness)

Young's Double-Slit Experiment

Thomas Young's famous experiment in the early 1800s provided crucial evidence for the wave nature of light, challenging the prevailing particle theory. 🔬

• Young directed a beam of light through two closely spaced slits and observed an interference pattern on a distant screen
• The alternating bright and dark fringes could only be explained if light behaved as a wave
• This experiment became a cornerstone of wave optics and helped establish the wave theory of light
• The mathematical relationships derived from this experiment allow us to determine the wavelength of light by measuring the fringe spacing

Visual Representations of Patterns

The visual patterns created by double-slit interference contain valuable information about both the light and the experimental setup.

• The spacing between bright fringes is directly proportional to wavelength and inversely proportional to slit separation
• The width of the central maximum depends on the width of individual slits
• By measuring the fringe pattern, we can determine:
  • The wavelength of the light (if slit separation is known)
  • The slit separation (if wavelength is known)
  • The relative phase of the light waves

Diffraction Gratings

A diffraction grating extends the double-slit concept to many evenly spaced parallel slits or lines, creating sharper and more distinct interference patterns.

• Modern gratings can contain thousands of slits per millimeter, producing very sharp spectral lines

• The basic equation for diffraction gratings is similar to the double-slit formula: $d \sin \theta = m\lambda$

  Where:

  • d is the distance between adjacent slits
  • θ is the angle to the mth order maximum
  • m is the order number (0, 1, 2, etc.)
  • λ is the wavelength

• Practical applications of diffraction gratings include:

  • Spectroscopy for identifying elements by their characteristic emission spectra
  • Wavelength calibration in optical instruments
  • Telecommunications for multiplexing different wavelengths

White Light and Diffraction Gratings

When white light (containing all visible wavelengths) passes through a diffraction grating, it creates a spectacular display of color. 🌈

• The central maximum (m = 0) appears white because all wavelengths constructively interfere at θ = 0°
• Higher-order maxima (m = 1, 2, etc.) spread into rainbow-like spectra because:
  • Different wavelengths diffract at different angles according to $d \sin \theta = m\lambda$
  • Red light (longer wavelength) diffracts at larger angles than violet light (shorter wavelength)
  • This creates a spectrum with red farthest from center and violet closest to center
• The visible spectrum appears in the order ROYGBIV (red, orange, yellow, green, blue, indigo, violet)
• Each successive order (m = 1, 2, 3...) produces a wider spectrum, with higher orders potentially overlapping

Practice Problem 1: Double-Slit Interference

Solution

To find the position of the third-order bright fringe, we use the equation: $y_{max} = \frac{m\lambda L}{d}$

Given:

• Wavelength (λ) = 650 nm = 650 × 10⁻⁹ m
• Slit separation (d) = 0.15 mm = 0.15 × 10⁻³ m
• Distance to screen (L) = 2.0 m
• Order number (m) = 3

Substituting these values: $y_{max} = \frac{3 \times (650 \times 10^{-9}) \times 2.0}{0.15 \times 10^{-3}}$ $y_{max} = \frac{3 \times 650 \times 10^{-9} \times 2.0}{0.15 \times 10^{-3}}$ $y_{max} = \frac{3.9 \times 10^{-6}}{0.15 \times 10^{-3}}$ $y_{max} = 26 \times 10^{-3} \text{ m} = 26 \text{ mm}$

Therefore, the third-order bright fringe is located 26 mm from the central maximum.

Practice Problem 2: Diffraction Grating

Solution

To find the wavelength, we'll use the diffraction grating equation: $d \sin \theta = m\lambda$

First, we need to find the slit separation (d):

• The grating has 5000 lines per centimeter
• So d = (1 cm) ÷ 5000 = 0.0002 cm = 2 × 10⁻⁶ m

Given:

• Slit separation (d) = 2 × 10⁻⁶ m
• Angle to first-order maximum (θ) = 15.5°
• Order number (m) = 1

Rearranging the equation to solve for λ: $\lambda = \frac{d \sin \theta}{m}$

Substituting the values: $\lambda = \frac{2 \times 10^{-6} \times \sin 15.5°}{1}$ $\lambda = 2 \times 10^{-6} \times 0.267$ $\lambda = 5.34 \times 10^{-7} \text{ m} = 534 \text{ nm}$

Therefore, the wavelength of the light is 534 nm, which is in the green region of the visible spectrum.