Light Waves

Optical path is the equivalent distance a light wave would travel in vacuum to accumulate the same phase change as it does when traveling through a medium.

For a medium with refractive index $\mu$ and physical distance $\Delta x$:

• Optical path = $\mu \cdot \Delta x$
• Phase change: $\delta = \frac{\omega}{c} \cdot \mu \cdot \Delta x = \frac{2\pi}{\lambda_0} \cdot \mu \cdot \Delta x$

Significance:

• Provides a unified way to calculate phase differences
• Explains why light appears to bend at interfaces
• Essential for analyzing interference in non-vacuum media
• Used to design optical instruments with specific path differences
• Explains why optical thickness differs from physical thickness

Example: A 1 mm glass plate ($\mu = 1.5$) has an optical path of 1.5 mm.

According to Huygens' Principle, reflection follows these steps:

1. When a plane wavefront (AB) reaches a reflecting surface (σ), each point on the surface becomes a secondary source of wavelets
2. These wavelets propagate back into the incident medium as hemispheres with radii proportional to time elapsed
3. The envelope of these secondary wavelets forms the reflected wavefront (CD)
4. The angle of incidence equals the angle of reflection

Mathematical relationship:

• If a wavelet from point A expands with radius vt (where v is light speed)
• And point C is just being reached by the incident wavefront
• Then the reflected wavefront is the tangent plane CD touching the hemisphere from A

This confirms the law of reflection without requiring the corpuscular theory of light.

Rayleigh Scattering:

• For particles smaller than wavelength, scattering intensity ∝ 1/λ⁴
• Shorter wavelengths (blue/violet) scatter much more than longer wavelengths (red)

Natural phenomena explained:

1. Blue sky:

   • Air molecules scatter blue light more strongly
   • Scattered blue light reaches our eyes from all directions

2. Red sunset/sunrise:

   • Sunlight travels longer path through atmosphere
   • Blue components scattered away
   • Remaining light appears red-orange

3. Clear vs. hazy sky:

   • Pure air molecules → deep blue sky
   • Water droplets/dust (larger particles) → lighter blue or grayish
   • Industrial pollution creates hazy appearance

Example: Red light (650nm) scatters (650/450)⁴ ≈ 4.3 times less than blue light (450nm), allowing red emergency signals to travel farther

• Optical path = geometric path × refractive index
• The ray with the longest geometric path traverses less distance through the high-refractive-index lens material
• The ray through the center travels a shorter geometric path but passes through more lens material (higher refractive index)
• The optical path difference (OPD) between any two rays equals zero because:
  • Longer geometric path × less time in high-index material = Shorter geometric path × more time in high-index material
• This principle ensures constructive interference at the focal point
• This is a fundamental principle in optical design that enables lenses to form coherent images

The phase change during reflection affects thin film interference in these ways:

• When light reflects from air-to-film interface: 180° phase change occurs
• When light reflects from film-to-substrate interface: phase change depends on relative refractive indices
• These phase shifts modify the conditions for constructive/destructive interference:
  • For reflection viewing: Maximum brightness occurs when 2μd = (n+1/2)λ
  • For transmission viewing: Maximum brightness occurs when 2μd = nλ

This principle explains why:

• Oil films on water display colorful patterns
• Soap bubbles show rainbow colors
• Anti-reflection coatings work on lenses

The first minimum (dark fringe) in Fraunhofer diffraction by a single slit occurs when:

$b\sin\theta = \lambda$

Where:

• b = width of the slit
• θ = angle of diffraction from the central maximum
• λ = wavelength of light

This occurs because the path difference between light from the top edge and the middle of the slit equals λ/2, causing destructive interference.

In general, minima (dark fringes) occur at angles where: $b\sin\theta = n\lambda$ where n = 1, 2, 3... (non-zero integers)

• Definition: Fresnel diffraction at a straight edge occurs when light waves pass by a sharp opaque obstacle and the observer's screen is at a finite distance from the diffracting edge
• Key characteristics:
  • Light intensity gradually decreases inside the geometric shadow region
  • Above the shadow boundary, intensity shows alternating maxima and minima
  • These alternations diminish in contrast as distance from shadow boundary increases
  • Eventually uniform illumination is reached far from the edge
• Mathematical relationship: The pattern depends on Fresnel integrals that involve path differences from various points on the wavefront
• Real-world application: This effect can be observed when light passes the edge of a razor blade in a darkened room

• Geometric optics prediction: Sharp boundary between fully illuminated region and complete shadow
• Actual wave optics result:
  • No sharp boundary exists
  • Light penetrates into the geometric shadow region
  • Intensity oscillates above the geometric shadow boundary
  • Maximum intensity can exceed that of the incident light (≈1.37 times)
• Significance:
  • Demonstrates wave nature of light
  • Proves geometric optics is an approximation valid only when λ ≪ object dimensions
  • Provides experimental evidence for Huygens' Principle
  • Shows diffraction effects are inherent to all wave phenomena
• Practical importance: Must be considered in precision optical instruments where sharp edges are present
• Historical impact: Helped resolve the wave-particle debate in favor of wave theory (before quantum mechanics)

The Rayleigh Criterion states that two point sources are considered "just resolved" when:

• The central maximum of one diffraction pattern falls on the first minimum of the other • The separation between centers equals the radius of the Airy disc • The intensity distribution shows a distinguishable dip (approximately 26.5% lower) between the two maxima

For a circular aperture, this occurs when the angular separation equals: $\theta = 1.22 \frac{\lambda}{D}$

Where λ is wavelength and D is aperture diameter.

Note: This represents the theoretical resolution limit of optical instruments due to diffraction effects.

When δ = π/2 and E₁ = E₂:

• The electric field components are:
  • E_y = E₁ sin(ωt - kx + π/2)
  • E_z = E₁ sin(ωt - kx)
• This creates tanθ = tan(ωt - kx), meaning θ = ωt - kx
• The electric field vector rotates at uniform angular speed ω
• The magnitude |E| = E₁ (remains constant)
• The tip of the electric field traces a perfect circle
• This represents circularly polarized light