Diffraction & Interference of Light

Fundamental Concept: Diffraction

  • Diffraction = spreading of light when it encounters an opening/obstacle whose size is on the order of the wavelength.
  • Evidence that light behaves as a wave (contrasts with straight-line ray optics).
  • Qualitative trends
    • Narrower slit ⇒ greater spreading.
    • Obstacles/openings ≫ \lambda behave almost like sharp edges (minimal diffraction).
  • Practical reminder: ordinary non-refracted light is often approximated as straight, but diffraction invalidates this for small apertures.

Single-Slit Diffraction (Slit–Lens System)

  • Experimental set-up: narrow slit → converging lens → viewing screen.
  • Observed pattern
    • Bright central fringe (central maximum) flanked by alternating dark and bright fringes.
    • Central maximum is twice as wide as each adjacent bright fringe.
    • As the slit width a decreases, the central maximum widens.
  • Condition for minima (dark fringes)
    • a \sin \theta = n\lambda
    • a = physical width of the slit.
    • \theta = angle between optic axis (normal) and line from lens center to the dark fringe.
    • n = integer (±1, ±2, …) that labels the dark fringe order.
    • \lambda = wavelength of incident light.
    • Bright fringes are located halfway between successive dark fringes.
  • Implications
    • Provides quantitative tool to measure wavelength or slit width.
    • Foundation for more complex multi-slit phenomena.

Multiple Slits & Young’s Double-Slit Experiment

  • Historical note: Thomas Young’s 1801 experiment cemented the wave theory of light.
  • Set-up: two parallel slits (separation d) illuminated with monochromatic light; screen placed at distance D.
  • Interference principle: overlapping wavefronts add algebraically (Ch. 7 review → superposition).
    • Constructive interference ⇒ bright fringes (maxima).
    • Destructive interference ⇒ dark fringes (minima).
  • Condition for dark fringes in Young’s arrangement
    • d \sin \theta = \left(n + \frac12\right)\lambda
    • Same symbol definitions as above, with d now the slit separation.
  • Condition for bright fringes
    • Mid-points between successive dark fringes.
  • Small-angle approximation (for \theta \lesssim 10^{\circ}):
    • \sin\theta \approx \tan\theta \approx \dfrac{y}{D} where y = linear fringe displacement on the screen.

Worked Example: Distance Between 6th and 8th Minima

Given: \lambda = 550\,{\rm nm}=5.50\times10^{-7}\,\text{m}, d = 0.14\,\text{mm}=1.4\times10^{-4}\,\text{m}, D = 70\,\text{cm}=0.70\,\text{m}.

  1. Dark-fringe position (small angle):
    y_n \approx \left(n + \tfrac12\right)\dfrac{\lambda D}{d}
  2. Required difference between 8th (n = 8) and 6th (n = 6) minima:
    \Delta y = y8 - y6
    \Delta y \approx \left[(8+\tfrac12) - (6+\tfrac12)\right]\dfrac{\lambda D}{d} = 2\,\dfrac{\lambda D}{d}
  3. Plug in numbers:
    \Delta y = 2 \times (5.50\times10^{-7}) \times 0.70 \,\big/\, (1.4\times10^{-4})
    = 5.5 \times 10^{-3}\,\text{m} = 5.5\,\text{mm}

Diffraction Gratings & Thin-Film Interference

  • Diffraction grating = large array of equally spaced slits.
    • Creates very sharp, wavelength-dependent maxima → acts like a prism but via interference.
    • Everyday example: closely spaced grooves on a CD/DVD produce iridescent rainbow.
  • Thin films (soap bubble, oil slick)
    • Two reflected rays (external vs. internal surface) interfere.
    • Depends on film thickness, wavelength, and phase shift upon reflection.
    • Note: interference here is from reflection, not diffraction, yet follows same superposition rules.
  • Practical relevance
    • Anti-reflective coatings in optics.
    • Structural coloration in nature (butterfly wings, beetle shells).

X-Ray Diffraction & Crystallography

  • X-ray wavelengths (~0.1 nm) match inter-atomic spacings; crystals act as 3-D diffraction gratings.
  • Technique: direct x-ray beam at crystal → record complex 2-D diffraction pattern of spots.
  • Computationally back-transform pattern to obtain electron-density map ⇒ deduce molecular structure (e.g., proteins, DNA).
  • Patterns are not linear fringes; they form concentric and radial spot arrays whose geometry encodes lattice spacing & symmetry.

Connections, Significance, & Implications

  • Wave phenomena (diffraction/interference) unify optics with general wave physics (Ch. 7 superposition).
  • Experimental confirmation of wave theory historically important; later quantum mechanics shows particles (electrons) diffract too.
  • Practical applications:
    • Optical instrumentation (spectrometers, lasers, holography).
    • Structural biology (x-ray crystallography, drug design) – ethical duty to handle sensitive biomedical data responsibly.
    • Engineering of anti-reflective & interference filters (smartphones, cameras).

Key Equations Quick Reference

  • Single-slit minima: a\sin\theta = n\lambda
  • Double-slit minima: d\sin\theta = \left(n + \tfrac12\right)\lambda
  • Small-angle link: \sin\theta \approx \tan\theta \approx \dfrac{y}{D}
  • Linear fringe spacing in double-slit (adjacent dark fringes): \Delta y = \dfrac{\lambda D}{d}
  • General constructive interference (multi-slit): d\sin\theta = m\lambda\quad(m=0,\pm1,\pm2,\dots)
  • Bragg’s law for x-ray diffraction (reflection viewpoint): 2d\sin\theta = m\lambda

Study Tips

  • Memorize the condition formulas and know when to swap a vs. d.
  • Practice small-angle approximations—it simplifies almost every exam calculation.
  • Conceptual cue: "Minima have the extra \tfrac12\lambda path difference." Bright fringes sit halfway.
  • Sketch typical patterns: single-slit (broad central) vs. double-slit (even spacing) to anchor qualitative reasoning.
  • Link ideas: grating → many slits = sharper peaks, thin film → reflection variant of 2-slit, x-ray → atomic-scale grating.