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.

Effects of changing parameters:

1. Increasing wavelength (λ):

   • Wider fringe spacing (w ∝ λ)
   • Reason: Longer wavelengths create larger phase differences for the same path length difference

2. Increasing slit separation (d):

   • Narrower fringe spacing (w ∝ 1/d)
   • Reason: With closer slits, a given angle corresponds to smaller path difference
   • When d >> λ, fringes become so narrow they may not be distinguishable

3. Increasing screen distance (D):

   • Wider fringe spacing (w ∝ D)
   • Reason: Same angular separation results in greater linear distance on a farther screen
   • This is why interference patterns are often projected on distant screens

Practical application: When using Young's experiment to measure an unknown wavelength, maximize D and minimize d to create wider, more easily measurable fringes.

Mathematical relationship: w = Dλ/d

Coherent Light Sources:

• Definition: Sources where the phase difference between emitted waves remains constant over time
• Requirements for interference: Sources must be coherent or derived from same primary source

Ordinary Light Sources (incoherent):

• Emit light in discrete wavetrains (length ~several meters)
• Active emission time ~10⁻⁸ seconds
• Random phase relationships between wavetrains
• Broad spectrum (not strictly monochromatic)
• Example: Sodium lamp has ~0.01nm linewidth

Coherent Sources:

• Maintain fixed phase relationships
• Emit long continuous wavetrains
• Narrower spectral width (more monochromatic)
• Example: Lasers emit wavetrains hundreds of meters long

In interference experiments:

• Young's double slit uses single source illuminating two slits to ensure coherence
• Path difference must be small (<few cm) for ordinary sources
• Laser sources can produce interference with path differences of several meters

• The wave undergoes a 180° (π radians) phase change upon reflection
• The reflected pulse is inverted compared to the incident pulse
• This applies to all types of waves including:
  • Mechanical waves on strings (when reflecting from heavier strings)
  • Light waves reflecting from optically denser media (higher refractive index)
  • Sound waves reflecting from denser media

Example: When light traveling in air reflects from glass or water, the reflected wave experiences a phase shift of π, which affects interference patterns in thin films.

A wave maintains its original phase upon reflection when:

• It reflects from a boundary where the transmission medium has lower density or lower wave impedance
• For mechanical waves: when a pulse reflects from a lighter string
• For light waves: when reflecting from a medium with lower refractive index
• For sound waves: when reflecting from a less dense medium

Example: Light traveling inside water that reflects from the water-air interface maintains its original phase because it's moving from higher refractive index (water) to lower refractive index (air).

• For a Fraunhofer diffraction setup with a second lens of focal length f:
• The position x on the observation screen relates to diffraction angle θ by:
  • x = f·sin(θ)
• For small angles (which is common in diffraction): x ≈ f·θ
• This means positions on the screen directly map to diffraction angles
• For a single slit of width a, the first minimum occurs at:
  • sin(θ) = λ/a
  • Position on screen: x = f·λ/a
• This relationship allows direct measurement of diffraction patterns and easy calculation of feature sizes or wavelengths
• Example: For slit width 0.1 mm, wavelength 500 nm, and focal length 50 cm, the first minimum appears at x = 2.5 mm from center

Secondary maxima decrease rapidly in intensity because:

1. The mathematical form of the intensity function $I = I_0 \frac{\sin^2 \beta}{\beta^2}$ causes rapid decline as β increases

2. Quantitatively:

   • First secondary maximum: only ~4.5% ($I_0/22$) of central maximum intensity
   • Second secondary maximum: only ~1.6% ($I_0/62.5$) of central maximum
   • Further maxima continue this declining trend

3. Physical explanation:

   • Secondary maxima result from partial constructive interference
   • At these positions, only some portions of the wavefront contribute constructively
   • Most path differences from different parts of the slit cause partial cancellation
   • Central maximum is unique because all parts of the wavefront are in phase

This rapid intensity decrease explains why practical applications focus mainly on the central maximum, as secondary maxima contribute minimally to the overall diffraction pattern.

Intensity distribution for two point sources at different resolution states:

1. Unresolved sources: • Single merged intensity peak • No discernible separation between sources • Peak intensity higher than individual source intensities

2. Just resolved sources (Rayleigh criterion): • Two peaks with a dip of ~26.5% between them • The mathematical condition: separation equals radius of Airy disk • Angular separation θ = 1.22λ/D

3. Well resolved sources: • Two distinct peaks with deep valley between them • Minimal interference between diffraction patterns • Intensity distribution closely matches two isolated point sources

The transition is defined by the mathematical relationship: θ = 1.22λ/D (for circular apertures)

The resulting intensity pattern for two equal point sources follows: I(θ) = I₁(θ) + I₂(θ) + 2√(I₁(θ)I₂(θ))cos(Δφ) where I₁ and I₂ are individual Airy patterns and Δφ is the phase difference.

• Unresolved images:

• Show a single merged intensity peak
• No visible separation between maxima
• Impossible to distinguish as separate objects
• Combined intensity profile is wider than a single object

• Just resolved images (Rayleigh limit):

• Show two maxima with a small dip (~26.5% intensity drop) between them
• Central maximum of one pattern falls on first minimum of the other
• Separation = radius of Airy disc

• Well-resolved images:

• Show two clearly separated intensity peaks
• Deep valley between maxima
• Minimal overlap between diffraction patterns
• Each object can be distinctly identified

Law of Malus:

• When polarized light with E-vector at angle θ to transmission axis passes through a polaroid:
• The transmitted component is E cos(θ)
• Since intensity I ∝ E², the transmitted intensity is: I = I₀ cos²θ
• Where I₀ is the intensity when the incident E-vector is parallel to the transmission axis
• Example: At θ = 45°, the transmitted intensity is 50% of the incident intensity