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.

The historical development of light's wave-particle duality:

Newton's Particle Theory (1600s):

• Light consists of tiny corpuscles
• Explained rectilinear propagation and shadows
• Successfully described reflection via elastic collisions
• Incorrectly predicted higher speed in denser media

Huygens' Wave Theory (1690s):

• Light as a wave phenomenon
• Initially overlooked due to Newton's authority

Wave Theory Confirmation (1800s):

• Young's double-slit experiment (1801) demonstrated interference
• Fresnel's diffraction experiments
• Maxwell's electromagnetic theory (1860s)
• Foucault's measurement showing light travels slower in water

Quantum Revolution (1900s):

• Photoelectric effect couldn't be explained by wave theory
• Einstein proposed light quanta (photons) in 1905

Modern Understanding:

• Light exhibits both wave and particle properties
• Behavior depends on the type of interaction being observed
• Wave-particle duality extends to all matter (electrons, atoms, etc.)

• Calculate the optical path length (OPL) for each path using: OPL = ∑(n_i × d_i)
  • Where n_i is the refractive index of each medium
  • d_i is the geometric distance traveled in each medium
• Determine the optical path difference (OPD) between the paths
• Calculate the phase difference: Δϕ = (2π/λ₀) × OPD
  • Where λ₀ is the wavelength in vacuum
• Same phase shift occurs when:
  • OPD = 0 (zero path difference)
  • OPD = mλ₀ (where m is an integer)
• Example: A ray traveling 10mm in air (n=1) and 2mm in glass (n=1.5) has the same OPL as a ray traveling 7mm in air and 3mm in glass (10×1 + 2×1.5 = 7×1 + 3×1.5 = 13)

• 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

When light travels through a medium with refractive index μ:

• The phase changes by δ = (ω/c)·(μΔx) in the medium
• The same phase change would require a distance of μΔx in vacuum
• This relationship defines the optical path length: a path length of Δx in a medium is equivalent to μΔx in vacuum

Example: Light traveling through 2mm of glass (μ = 1.5) experiences the same phase change as light traveling through 3mm of vacuum.

Note: This explains why interference patterns depend on optical path difference, not just geometric path difference.

To calculate the minimum thickness for constructive interference in reflected light:

1. Use the formula: 2μd = (n+1/2)λ, where n=0 for minimum thickness

2. Therefore: 2μd = λ/2

3. Solving for d: d = λ/(4μ)

4. Step-by-step calculation example:

   • Given: λ = 500 nm, μ = 1.33 (water)
   • d = 500 nm/(4×1.33) = 94 nm

This minimum thickness creates a path difference that, combined with the π phase shift at the upper surface, produces constructive interference.

Note: For destructive interference in reflection, the minimum thickness would be d = λ/(2μ).

Diffraction patterns form based on Huygens' Principle, which states:

• Every point on a wavefront acts as a source of secondary spherical wavelets
• When a wavefront is partially obstructed, only wavelets from exposed parts superpose
• These wavelets interfere constructively and destructively, creating patterns of varying intensity

This contradicts the simple ray model of light in several ways:

• Ray optics predicts sharp shadows with uniform darkness
• Diffraction shows light bending around corners into geometric shadow regions
• Intensity in diffraction patterns varies periodically rather than having clear boundaries
• The pattern complexity depends on wavelength, which ray optics doesn't account for

The ability of light to bend around obstacles and create interference patterns provides direct evidence of its wave nature, something the ray model cannot explain.

Example: When a laser beam passes through a small circular aperture, rather than projecting a simple circle of light, it produces a pattern of concentric bright and dark rings (Airy disk pattern).

The radius R of the first dark ring on a screen placed at distance D from a circular aperture is:

$R \approx 1.22 \frac{\lambda D}{b}$

Where:

• λ is the wavelength of light
• D is the distance from aperture to screen
• b is the diameter of the circular aperture

This radius defines the boundary of the central bright disc (known as the Airy disc) in the diffraction pattern.

Note: This formula assumes Fraunhofer diffraction conditions where D is much larger than b.

Physical meaning of phase difference δ:

• δ represents the temporal offset between oscillations in orthogonal directions
• It determines how the electric field vector's tip traces its path in space

Effects on polarization state:

• δ = 0: Components reach maxima simultaneously → linearly polarized along a line with slope E₂/E₁
• δ = π: Components reach maxima in opposition → linearly polarized along a line with slope -E₂/E₁
• δ = π/2, E₁ = E₂: Circular polarization → electric field traces a circle
• δ = π/2, E₁ ≠ E₂: Elliptical polarization → electric field traces an ellipse
• δ = random (changing with time): Unpolarized light → electric field direction changes randomly
• δ = any other fixed value: Elliptical polarization with different orientation and eccentricity

The phase difference essentially encodes the "shape" of the path traced by the electric field vector.

When linearly polarized light meets a perpendicular polaroid (θ = 90°):

Mathematical analysis:

• By Law of Malus: I = I₀cos²(90°) = I₀ × 0 = 0
• The electric field component along transmission axis is E×cos(90°) = 0

Physical explanation:

• The incident E-vector has no component along the transmission axis
• The electric field is entirely parallel to the blocking chains in the polaroid
• All the electric field energy induces currents in these chains
• These currents dissipate energy as heat in the polaroid material
• No electromagnetic energy is transmitted through the polaroid
• This creates complete extinction of the light (100% blocking)

This principle is utilized in applications like LCD screens and glare-reducing polarized sunglasses.