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:
   $\Delta y = y<em>8 - y</em>6$
   $\Delta y \approx \left[(8+\tfrac12) - (6+\tfrac12)\right]\dfrac{\lambda D}{d} = 2\,\dfrac{\lambda D}{d}$
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.