Two-beam interference is a key concept in optics. It occurs when two light waves combine, creating patterns of bright and dark regions. Understanding the conditions and calculations for interference patterns is crucial for grasping how light behaves in various optical systems.

Multiple-beam interference takes this concept further, involving reflections between parallel surfaces. This phenomenon leads to sharper fringes and higher peak intensities, making it valuable for applications requiring precise measurements or improved signal detection in optical devices.

Fundamentals of Two-Beam Interference

Conditions for two-beam interference

• Coherence ensures light sources have fixed phase relationship
  • Temporal coherence maintains phase over time
  • Spatial coherence maintains phase across wavefront
• Polarization states of interfering waves must match (linear, circular, elliptical)
  • Maximized interference when polarization is parallel
• Path difference determines phase difference between waves
  • Constructive interference: Path difference is integer multiple of wavelength ($n\lambda$)
  • Destructive interference: Path difference is half-integer multiple of wavelength ($(n+\frac{1}{2})\lambda$)

Calculations for interference patterns

• Intensity distribution depends on individual wave intensities ($I_1$, $I_2$) and phase difference ($\delta$)
  • Formula: $I = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos(\delta)$
  • Phase difference relates to path difference ($\Delta L$): $\delta = \frac{2\pi}{\lambda} \Delta L$
• Fringe spacing is distance between adjacent bright or dark fringes
  • Double slits (separation $d$, observation distance $L$): $\Delta y = \frac{\lambda L}{d}$
  • Thin films: Spacing depends on film thickness, wavelength, and incidence angle

Principles and Applications of Multiple-Beam Interference

Principles of multiple-beam interference

• Multiple reflections between parallel surfaces (Fabry-Perot interferometer) contribute to interference pattern
• Advantages over two-beam interference:
  1. Sharper fringes as number of interfering beams increases
  2. Higher peak intensity improves signal-to-noise ratio and detection sensitivity

Analysis of multiple-beam patterns

• Intensity distribution follows Airy function: $I = \frac{I_0}{1 + F \sin^2(\delta/2)}$
  • $I_0$: Peak intensity
  • $F$: Coefficient of finesse (depends on surface reflectivity)
  • $\delta$: Phase difference between reflections, relates to surface separation ($d$), wavelength ($\lambda$), and incidence angle ($\theta$): $\delta = \frac{4\pi}{\lambda} d \cos(\theta)$
• Resolution determined by finesse ($\mathcal{F}$), ratio of free spectral range (FSR) to full width at half maximum (FWHM) of peaks
  • Formula: $\mathcal{F} = \frac{\text{FSR}}{\text{FWHM}}$
  • Higher finesse enables distinguishing closely spaced wavelengths