Geometric and wave optics: Snell's law, Fresnel equations, polarization, diffraction, interference, thin-lens equation. Classical-regime light propagation, complementary to EM (field equations) and wave-phenomena (generic wave theory).
optics
Snell's law
At an interface between media with indices n₁, n₂ : n₁ sin θ₁ = n₂ sin θ₂.
Fresnel equations
Reflection and transmission amplitude coefficients at a dielectric interface, split into s- and p-polarizations. Give reflectance,…
Polarization of light
Direction of oscillation of the electric-field vector. Linear, circular, elliptical. Basis for polarizers, LCD displays, and optical…
Diffraction
Bending of waves around obstacles or through apertures, producing characteristic intensity patterns. Airy disk sets the diffraction limit…
Optical interference
Superposition of coherent waves produces fringes (constructive/destructive addition). Young's double-slit is the canonical demonstration;…
Thin-lens equation
1/f = 1/d_o + 1/d_i. Image formation by a thin lens in the paraxial (small-angle) regime.
Gaussian beam
Paraxial-optics Gaussian solution to the Helmholtz equation: waist w_0, Rayleigh range z_R = π w_0²/λ, hyperbolic divergence. Fundamental…
Fraunhofer diffraction
Far-field diffraction pattern equal to the Fourier transform of the aperture function. Airy disk for a circular aperture; sinc pattern for…
Fresnel diffraction
Near-field diffraction integral; retains the quadratic phase in the aperture; bridges geometrical and Fraunhofer regimes.
Laser (stimulated emission)
Amplification of light by stimulated emission in a population-inverted medium within a resonant cavity; produces coherent monochromatic…
Mode-locked lasers (femtosecond)
Phase-locked longitudinal modes synthesize ultrashort pulses (fs). Ti:sapphire ~10 fs; Kerr-lens ML. Enabled 2023 Nobel for attosecond…
Optical frequency comb
Mode-locked laser provides Δfrep-spaced lines spanning octave; self-referenced via f-2f. Hall/Hänsch Nobel 2005. Precision spectroscopy,…
Nonlinear optics (χ⁽²⁾, χ⁽³⁾)
Intensity-dependent polarization: SHG, SFG, OPA (χ⁽²⁾); self-phase modulation, Kerr, FWM (χ⁽³⁾). Enables frequency conversion, squeezed…
Plasmonics & SPR
Collective electron oscillations at metal-dielectric interface (surface plasmon polaritons) or nanoparticles (LSPR). Sub-λ confinement;…
Abbe diffraction limit
Min resolvable distance d = λ/(2 NA) with NA = n sin α; overcome by super-resolution (STED, STORM, PALM) via fluorescence switching.
Point-spread function (PSF)
Image of point source; Airy disk for circular aperture; convolution kernel; Strehl ratio measures wavefront quality.
Modulation transfer function (MTF)
Fourier transform of PSF magnitude; measures contrast transfer vs spatial frequency; system performance metric.
ABCD ray matrix method
2×2 matrices compose ray transfer through lenses, mirrors, media; Gaussian beam q(z) transforms as q' = (Aq+B)/(Cq+D).
Optical cavity stability
0 ≤ (1-L/R₁)(1-L/R₂) ≤ 1 for stable confocal resonators; confocal, hemispherical, planar limits; finesse ℱ = π√R/(1-R).
Laser rate equations & threshold
dN/dt = R - N/τ - σcNφ; dφ/dt = cσNφ - φ/τ_c + spontaneous; threshold population inversion N_th = 1/(σcτ_c).
Q-switching & mode-locking
Q-switch dumps cavity energy in short giant pulse (ns, MW); mode-locking phase-locks longitudinal modes giving fs-ps pulses; Kerr-lens ML.
Chirped-pulse amplification
Stretch pulse in time, amplify, recompress; avoids nonlinear damage; enabled PW lasers; 2018 Nobel (Mourou, Strickland).
Optical frequency comb
Mode-locked laser produces equally-spaced spectral lines f_n = n f_rep + f_ceo; connects RF to optical frequencies; 2005 Nobel (Hall,…
Photonic bandgap crystals
Periodic dielectric forbids EM propagation in frequency bands; analogous to electronic bandgap; inhibits spontaneous emission.
Negative-index metamaterials
ε<0, μ<0 simultaneously give n<0; reverse Doppler, Cherenkov; perfect lens of Pendry; cloaking via transformation optics.
Kerr effect (χ³)
n = n₀ + n₂ I; self-focusing, self-phase modulation; Kerr lens in Ti:Sapphire mode-locking; soliton propagation in fiber.
Second-harmonic generation (SHG)
χ² process converts 2ω₁ → ω₂; requires non-centrosymmetric crystal; phase matching Δk=0 via birefringence or quasi-phase-matching.
Spontaneous parametric down-conversion
ω_p → ω_s + ω_i in χ² crystal; source of entangled photon pairs; basis of quantum optics experiments, Bell tests.
Four-wave mixing (χ³)
ω₃ = ω₁+ω₂-ω₄ in isotropic media; generates squeezed light, optical phase conjugation; CARS spectroscopy.
Raman scattering
Inelastic photon-phonon scattering: Stokes ω-ω_v, anti-Stokes ω+ω_v; molecular fingerprinting; SERS enhancement factor 10⁶-10⁸.
Electro-optic Pockels effect
χ² response to DC field gives birefringence Δn ∝ E; LiNbO₃ modulators in telecom; GHz-bandwidth optical switches.
Acousto-optic modulation
Bragg diffraction off traveling sound wave; Doppler-shifts ω_out = ω_in ± Ω_sound; tunable deflectors, frequency shifters.
Faraday rotation
Plane polarization rotates by VBℓ in magnetic field; Verdet constant V; basis of optical isolators.