Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Electromagnetism of Continuous Media — Optics Petr Kužel Course for 3rd year of University studies (Université Paris-Nord) Series of 14 lectures in Department of Dielectrics, IOP Topics of Interest • • • • • • • • • Layered systems and optical coatings: Birefringence and polarization optics: Dispersion and pulse propagation: Electro- and magneto-optics: Nonlinear optics: Waveguides and fibers: Inhomogeneous media: Coherence: Diffraction, gaussian beams: in detail in detail yes yes basics basics basics marginal no Electromagnetic Spectrum 380 nm 770 nm visible spectrum RADIO WAVES OPTICAL RANGE γ-RAYS MW (radar) X-RAYS UV vacuum near wavelength (m) frequency: 10-16 10-14 10-12 3×1020 Hz UHF VHF FM AC AM IR near 10-10 10-8 (1 Å) (1 nm) 10-6 (1 µm) 3×1017 Hz 6×1014 Hz far 10-4 1 THz 10-2 (1 cm) 100 (1 m) 1 GHz 102 104 106 (1 km) 1 MHz 50 Hz 108 Notation conventions ×≡∧ curl E ≡ rot E ≡ ∇ × E ≡ ∇ ∧ E div E ≡ ∇ ⋅ E Maxwell Equations ∂B =0 ∂t ∂D = j ∇∧ H − ∂t ∇⋅ D = ρ ∇⋅B = 0 ∇∧E+ ∂ B ⋅ dS ∂t ∫ ∂ ⋅ = H l D ⋅ dS + J d ∫ ∂t ∫ ∫ E ⋅ dl = − ∫ D ⋅ dS = Q ∫ B ⋅ dS = 0 D = εE = ε 0 E + P B = µH = µ 0 H + M E 2t − E1t = 0 H 2t − H1t = js D2 n − D1n = σ B2 n − B1n = 0 Wave equation N 2 ∂2E ∇ E− 2 =0 2 c ∂t 2 N 2 ∂2H ∇ H− 2 =0 2 c ∂t 2 E = E0 ei (ωt − k ⋅r ) H = H 0 ei (ωt − k ⋅r ) Maxwell Equations ∂B =0 ∂t ∂D = j ∇∧ H − ∂t ∇⋅ D = ρ ∇⋅B = 0 ∇∧E+ Faraday (induction) E 2t − E1t = 0 Ampère (induction) H 2t − H1t = js Coulomb (electric charges) D2 n − D1n = σ No magnetic charges D = εE = ε 0 E + P B = µH = µ 0 H + M B2 n − B1n = 0 Discontinuity at the interface • Ideal conductor (σ = ∞) • Real conductor (σ < ∞) E E H H E 2t − E1t = 0 E 2t − E1t = 0 H 2t − H1t = js inhomogeneity H 2t − H1t = 0 Time-resolved reflectivity ref lec ted pro be • Pump pulse: high density of free carriers (non-equilibrium state) • Probe pulse: monitoring of the permittivity change mp u p ed y la de e b o pr Inhomogeneous distribution Properties of the permittivity • • • • • dispersive [ε = ε(ω)] — not dispersive (ε does not depend on ω) absorbing (ε is complex) — not absorbing or transparent (ε is real) inhomogeneous [ε = ε(r)] — homogeneous (ε does not depend on r) anisotropic (ε is a second rank tensor) — isotropic (ε is a scalar) nonlinear [ε = ε(E,H)] — linear (ε does not depend on the fields) D(ω) = ε(ω) E (ω) = ε 0 E (ω) + P (ω) P (ω) = ε 0 χ(ω) E (ω) ε(ω) = ε 0 (1 + χ(ω)) Spectral decomposition Real notation: • Monochromatic wave a (t ) = A cos(ωt + α ) = Re Aeiωt , • Polychromatic wave a (t ) = ∑ Ak cos(ωk t + α k ) { } with A = A eiα k ∞ a (t ) = ∫ A(ω) cos(ωt + α(ω)) dω 0 Complex notation: • Monochromatic wave a(t ) = Aeiωt ∞ • Polychromatic wave a(t ) = ∫ A(ω)e 0 iωt dω ( or a(t ) = or 1 a(t ) = ∫ A(ω)eiωt dω 2 −∞ 1 Aeiωt + A∗e −iωt 2 ∞ ) Calculation of the field products • If only mean values are needed (energy density and flow): a(t )b(t ) in complex notation: a (t )b(t ) T 1 1 = ∫ A cos(ωt + α ) B cos(ωt + β) = AB cos(α − β) 2 T0 1 = Re AB ∗ . 2 { } • If instantaneous values are needed (nonlinear optics): a(t ) = ( ) 1 Aeiωω+ A∗e −iωω 2 b(t ) = ( 1 Beiωt + B ∗e −iωt 2 ) Maxwell equations for the spectral components ∇ ∧ E = − iω B ∇ ∧ H = j + iωD ∇⋅ D = ρ ∇⋅B = 0 Introduction of the generalized permittivity: Total current (free + polarization charges): iσ jTOT = σE + iωD = iω ε − E = iωεˆ E ω We define a new D which involves the free charges: ∇ ∧ E = −iωB ∇ ∧ H = iωD ∇⋅ D = 0 ∇⋅B = 0 Dˆ = εˆ E Third Maxwell equation becomes: 1 ∇ ⋅ Dˆ = ∇ ⋅ (ε − i σ ω)E = ρ + ∇ ⋅ j = 0 iω Conservation of energy All conservation laws have the general form: ∂ρ +∇⋅ j = P ∂t Work of the field per unit of time exerted on the charges (Lorentz force): q(E + v ∧ B )⋅ v = qv ⋅ E → j ⋅ E From Maxwell equations: ∂B ∂t ∂D ∇∧H = j+ ∂t ∇∧E = − /⋅ H ∂D ∂B E H ⋅ + ⋅ + ∇ ⋅ (E ∧ H ) = − j ⋅ E ∂t ∂t /⋅ E ∂U +∇⋅S =−j⋅E ∂t Conservation of energy: continued E⋅ ∂D ∂B +H⋅ ∂t ∂t + ∇ ⋅ (E ∧ H ) = − j ⋅ E ∂U +∇⋅S =−j⋅E ∂t Linear non-dispersive (and non-absorbing) media: U = 12 (E ⋅ D + H ⋅ B ) Absorbing medium: for a monochromatic wave the energy change corresponds to the heat: Q= ( ) ( ) ω Im ε∗ EE ∗ = ω Im ε∗ E 2 2 Medium with small dispersion: quasi-monochromatic wave: 1 d (ωε ) 1 d (ωε ) 2 U = E ⋅ E ∗ + µ0 H ⋅ H ∗ = E + µ0 H 2 4 dω 2 dω