Light waves behave in fascinating ways when encountering obstacles or openings. The Huygens-Fresnel principle explains how waves spread and bend, creating intricate patterns of bright and dark spots. It's like ripples in a pond, but with light!

This principle is crucial for understanding diffraction, a key concept in optics. By treating each point on a wavefront as a source of new waves, we can predict how light will behave when it passes through small openings or around edges.

Huygens-Fresnel Principle and Wavefront Propagation

Huygens-Fresnel principle in diffraction

• Powerful tool for understanding and predicting behavior of light waves in presence of obstacles or apertures
  • Combines Huygens' principle each point on wavefront acts as secondary source of spherical wavelets with Fresnel's additions including concept of interference and inclination factor
• Wavefront at any later time determined by superposition of secondary wavelets
  • Wavelets interfere with each other resulting in observed diffraction patterns (interference fringes, bending of light around obstacles)
• Explains various diffraction phenomena
  • Bending of light around obstacles (diffraction around edges)
  • Formation of diffraction fringes (alternating bright and dark bands)
  • Spreading of light after passing through aperture (diffraction through slits or circular openings)

Wavefront propagation and diffraction

• Wavefront surface of constant phase in propagating wave
  • In light wavefronts perpendicular to direction of propagation represent surfaces of constant optical path length
• According to Huygens-Fresnel principle each point on wavefront acts as source of secondary spherical wavelets propagating outward with same speed as original wave
• Propagation of wavelets and their subsequent interference give rise to diffraction patterns observed when light encounters obstacles or apertures
  • Diffraction effects more pronounced when size of obstacle or aperture comparable to wavelength of light (small apertures, edges of objects)

Mathematical Formulation and Limitations

Mathematical formulation of Huygens-Fresnel

• Fresnel-Kirchhoff diffraction formula calculates complex amplitude of diffracted field at point $P$ $U(P) = \frac{1}{i\lambda} \iint_S U(Q) \frac{e^{ikr}}{r} \cos(\mathbf{n}, \mathbf{r}) dS$

  • $U(P)$ complex amplitude at point $P$
  • $U(Q)$ complex amplitude at point $Q$ on aperture
  • $\lambda$ wavelength of light
  • $k = \frac{2\pi}{\lambda}$ wave number
  • $r$ distance between points $P$ and $Q$
  • $\mathbf{n}$ normal vector to aperture surface
  • $\mathbf{r}$ vector from $Q$ to $P$
  • $dS$ infinitesimal area element on aperture surface

• Formula integrates contributions of secondary wavelets originating from each point on aperture

  • Takes into account amplitude, phase, and inclination factor $\cos(\mathbf{n}, \mathbf{r})$ of wavelets

Limitations of Huygens-Fresnel model

• Relies on several assumptions and approximations limiting accuracy in certain situations
  1. Assumes aperture or obstacle large compared to wavelength of light
  2. Assumes diffracted field observed at large distance from aperture (far-field approximation)
  3. Neglects vectorial nature of light treats it as scalar wave may lead to inaccuracies when dealing with polarization effects or highly focused beams
• Does not account for finite size of secondary wavelets can lead to discrepancies in predicted diffraction patterns especially in near-field region
• Despite limitations remains valuable tool for understanding and predicting diffraction phenomena in many practical situations
  • Particularly in far-field region and for apertures much larger than wavelength of light (visible light through macroscopic apertures)