Photonic crystals, waveguides, plasmons, and metamaterials allow unprecedented control over light at the wavelength scale and below. From optical fiber communications to single-photon emitters to cloaking devices, photonics bridges classical electrodynamics and quantum optics.
A photonic crystal is a periodic dielectric structure with period a ~ λ/2. By analogy with electron Bloch states in a crystal potential, photons in a periodic ε(r) form photonic bands. A photonic bandgap (PBG) is a range of frequencies for which no propagating modes exist.
The band structure ω_n(k) follows from this equation. Bandgap: between two bands, analogous to semiconductor bandgap. For a 1D Bragg stack (alternating n₁, n₂ layers of thickness d₁, d₂): maximum gap at λ = 2(n₁d₁ + n₂d₂) (quarter-wave condition). Gap width: Δω/ω₀ ≈ (4/π)|n₁−n₂|/(n₁+n₂).
Defect modes: a point defect in a 2D/3D photonic crystal creates localized resonant modes inside the bandgap — a photonic atom. A line defect creates a waveguide that routes light around sharp bends without loss. Applications: photonic crystal fibers (PCF), high-Q cavities (Q ~ 10⁶) for single-photon emitters, slow-light waveguides (v_g ≈ c/300).
Example PH.1 — Bragg Reflector and VCSEL Design
Design a GaAs/AlAs distributed Bragg reflector (DBR) for 850 nm emission. n_{GaAs} = 3.6, n_AlAs=3.0.Computethequarter−wavethickness,reflectivity,andthenumberofpairsfor R > 99%.
Reflectivity of N pairs (n₁/n₂ = AlAs/GaAs):ForNpairswithns=GaAssubstrate:RN=[(1−(n1/n2)2N(ns/n0))/(1+(n1/n2)(2N)(ns/n0))]2.Stop−bandcenter:highreflectivityregion.(n1/n2)2N=(3.0/3.6)(2N = 0.833^(2N).
Number of pairs for R = 99%:R>0.99requires(n1/n2)2N×(ns/n0)≫1.Forns=3.6,n0=1(air):(3.0/3.6)(2N)×3.6>threshold.TryN=20:(0.833)40×3.6=2.8×10−4×3.6≈10−3.R=(1−10−3)2/(1+10−3)2≈99.6emitting laser) uses two 20-pair DBRs with a half-wavelength GaAs active cavity — threshold current ~1 mA, used in fiber optic transceivers and LiDAR (iPhone).
Stopband width:Δλ/λ=(4/π)arcsin((n1−n2)/(n1+n2))≈(4/π)(n1−n2)/(n1+n2)=(4/π)(0.6/6.6)≈0.115.Δλ≈ 98 nm — very wide stopband useful for broadband reflectors.
PH.3 Plasmonics
Surface plasmon polaritons (SPPs): coupled oscillations of free electrons and EM field at a metal-dielectric interface. Dispersion relation:
For a Drude metal: ε_m(ω) = 1 − ω_p²/ω². The SPP wavevector k_SPP lies outside the light cone — SPPs are non-radiative (surface bound) until excited by evanescent coupling (prism coupling, grating coupler, or near-field tip).
Localized surface plasmons (LSP): collective electron oscillation in metallic nanoparticles. Mie theory resonance for a sphere of radius a ≪ λ: polarizability α = 4πa³(ε_m − ε_d)/(ε_m + 2ε_d) diverges at resonance ε_m = −2ε_d. For Au in water: resonance at λ ≈ 520 nm (gold is red/purple at nanoscale). Applications: SERS (surface-enhanced Raman, 10¹⁰× enhancement in nanogap), plasmonic sensors (binding shifts resonance), photothermal therapy.
PH.4 Metamaterials
Engineered structures with effective ε_eff and μ_eff not found in natural materials.Negative index material(NIM): ε < 0 and μ < 0 simultaneously → n = −√(εμ) < 0. Snell's law: n₁ sinθ₁ = n₂ sinθ₂ still holds but refraction is on the same side of the normal (negative refraction).
Perfect lens (Veselago-Pendry, 2000): a slab of n = −1 material focuses both propagating and evanescent waves — potentially diffraction-unlimited imaging (superlens). Experimental demonstrations at microwave, infrared.
Transformation optics: map desired light trajectories to a required ε(r) and μ(r). Pendry (2006) showed a cloak can guide light around an object: ε and μ must vary as functions of position (anisotropic, inhomogeneous). First microwave cloak demonstrated (Schurig et al. 2006, Science). At optical frequencies: material loss is a fundamental limitation.
Definition PH.2 — Common Traps
Single-mode depends on V-number: core size, wavelength, and numerical aperture all matter.
Group velocity is pulse velocity: phase velocity alone does not determine information transport.
Band gaps require periodicity on wavelength scale: ordinary index contrast is not enough.
Metamaterial losses are real constraints: negative index behavior can be overwhelmed by absorption.