Electrodynamics Lecture D Manfred Hammer, Hugo Hoekstra, Lantian Chang, Mustafa Sefünç, Yean-Sheng Yong Integrated Optical MicroSystems MESA + Institute for Nanotechnology University of Twente, The Netherlands University of Twente, Enschede, The Netherlands Course 1912104101), Block 1A, 2013/2014 Department of Electrical Engineering, University of Twente P.O. Box 217, 7500 AE Enschede, The Netherlands Phone: +31/53/489-3448 Fax: +31/53/489-3996 E-mail: m.hammer@utwente.nl 1
Course overview Electrodynamics A B C D E F G H Introduction, Maxwell equations, brush up on vector calculus, Dirac delta, potentials, Taylor expansion, Fourier transform, Maxwell equations, brush up on electro- and magnetostatics, multipole expansion Maxwell equations, microscopic and macroscopic, time- and frequency domain, differential and integral form, interface conditions, continuity equation, energy and momentum of electromagnetic fields Wave equation, plane waves, plane harmonic electromagnetic waves, refractive index, polarization, energy transport, spherical waves Reflection and transmission at interfaces, lossy materials, wave packets, dispersion, phase and group velocity Maxwell equations, vectorial and scalar Helmholtz equation, 2D configurations intermezzo), mode problems, metallic and dielectric waveguides Scalar and vector potentials, gauge conditions, retarded potentials, electric and magnetic dipole radiation Special relativity; transmission lines 2
Wave equation Homogeneous macroscopic Maxwell equations: ρ f = 0, J f = 0) D = 0, E = Ḃ, B = 0, H = Ḋ 3
Wave equation Homogeneous macroscopic Maxwell equations: ρ f = 0, J f = 0) D = 0, E = Ḃ, B = 0, H = Ḋ Linear isotropic homogeneous media: ǫr), µr)!) D = ǫ 0 ǫe, B = µ 0 µh 3
Wave equation Homogeneous macroscopic Maxwell equations: ρ f = 0, J f = 0) D = 0, E = Ḃ, B = 0, H = Ḋ Linear isotropic homogeneous media: ǫr), µr)!) D = ǫ 0 ǫe, B = µ 0 µh E = 0, E = µ 0 µḣ, H = 0, H = ǫ0ǫė, 3
Wave equation Homogeneous macroscopic Maxwell equations: ρ f = 0, J f = 0) D = 0, E = Ḃ, B = 0, H = Ḋ Linear isotropic homogeneous media: ǫr), µr)!) D = ǫ 0 ǫe, B = µ 0 µh E = 0, E = µ 0 µḣ, H = 0, H = ǫ0ǫė, E = ǫ 0 µ 0 ǫµë, H = ǫ 0µ 0 ǫµḧ. 3
Wave equation 1 2 ) t 2 E = 0, 1 2 ) t 2 H = 0, speed of light in vacuum: c = 1 ǫ0 µ 0, phase velocity in the medium: c m = 1 ǫ0 µ 0 ǫµ = c = c ǫµ n, refractive index of the medium: n = ǫµ. Shorter: E = 0, H = 0, = 1 2 ) t 2 : Quabla, 4
Wave equation 1 2 ) t 2 E = 0, 1 2 ) t 2 H = 0, speed of light in vacuum: c = 1 ǫ0 µ 0, phase velocity in the medium: c m = 1 ǫ0 µ 0 ǫµ = c = c ǫµ n, refractive index of the medium: n = ǫµ. Shorter: E = 0, H = 0, = 1 2 ) t 2 : Quabla, d Alembert operator). 4
Plane waves 1 2 ) t 2 ψr,t) = 0. 5
Plane waves 1 2 ) t 2 ψr,t) = 0. ) ψr,t) = fk r ωt)+bk r +ωt) solves ) for ω 2 = k 2, f, b arbitrary smooth), ω > 0, k = k ) 5
Plane waves 1 2 ) t 2 ψr,t) = 0. ) ψr,t) = fk r ωt)+bk r +ωt) solves ) for ω 2 = k 2, f, b arbitrary smooth), ω > 0, k = k ) Features off appear on planes with k r ωt = ϕ 0 const., i.e. move with velocity c m = ω/k in direction k/k. r k k = ϕ 0 k + ω k t 5
Plane waves 1 2 ) t 2 ψr,t) = 0. ) ψr,t) = fk r ωt)+bk r +ωt) solves ) for ω 2 = k 2, f, b arbitrary smooth), ω > 0, k = k ) Features off appear on planes with k r ωt = ϕ 0 const., i.e. move with velocity c m = ω/k in direction k/k. Features ofbappear on planes with k r +ωt = ϕ 1 const., i.e. move with velocity c m = ω/k in direction k/k. r k k = ϕ 0 k + ω k t r k k = ϕ 1 k ω k t 5
Plane waves 1 2 ) t 2 ψr,t) = 0. ) ψr,t) = fk r ωt)+bk r +ωt) solves ) for ω 2 = k 2, f, b arbitrary smooth), ω > 0, k = k ) Features off appear on planes with k r ωt = ϕ 0 const., i.e. move with velocity c m = ω/k in direction k/k. Features ofbappear on planes with k r +ωt = ϕ 1 const., i.e. move with velocity c m = ω/k in direction k/k. Planes k: wave fronts, phase fronts. r k k = ϕ 0 k + ω k t r k k = ϕ 1 k ω k t 5
Plane harmonic waves 1 2 ) t 2 ψr,t) = 0. 6
Plane harmonic waves 1 2 ) t 2 ψr,t) = 0. frequency domain: ψr,t) = 1 2 ψr) e iωt + c.c., ) + ω2 ψr) = 0, Helmholtz equation) 6
Plane harmonic waves 1 2 ) t 2 ψr,t) = 0. frequency domain: ψr,t) = 1 2 ψr) e iωt + c.c., ) + ω2 ψr) = 0, Helmholtz equation) spatial oscillations: ψr,t) = 1 2 ψ 0 e ik xx e ik y y e ik z z e iωt + c.c., ) k 2 + ω2 ψ 0 = 0, ψ 0 C, only ψ 0 0 is of interest, k = k x,k y,k z ) 6
Plane harmonic waves 1 2 ) t 2 ψr,t) = 0. frequency domain: ψr,t) = 1 2 ψr) e iωt + c.c., ) + ω2 ψr) = 0, Helmholtz equation) spatial oscillations: ψr,t) = 1 2 ψ 0 e ik xx e ik y y e ik z z e iωt + c.c., ) k 2 + ω2 ψ 0 = 0, ψ 0 C, only ψ 0 0 is of interest, k = k x,k y,k z ) ψr,t) = Reψ 0 e ik r ωt). 6
Plane harmonic waves ψr,t) = Reψ 0 e ik r ωt) Periodicity in time: angular frequency: ω, period: T = 2π ω, Spatial periodicity: wave vector: k, k = k, wavenumber: k = ω = ω c m c n = k 0n, Electromagnetic spectrum vacuum wavenumber: k 0 = ω c, refractive index: vacuum wavelength: wavelength in the medium: n = ǫµ, λ = 2π k 0 = 2πc ω, λ m = 2π k = 2π k 0 n = λ n. 7
Plane harmonic electromagnetic waves E = 0, E = µ 0 µḣ, H = 0, H = ǫ0ǫė, & Er,t) = E 0 e ik r ωt), Hr,t) = H 0 e ik r ωt) skip1/2, Re, c.c.) 8
Plane harmonic electromagnetic waves E = 0, E = µ 0 µḣ, H = 0, H = ǫ0ǫė, & Er,t) = E 0 e ik r ωt), Hr,t) = H 0 e ik r ωt) skip1/2, Re, c.c.) or Ẽ = 0, Ẽ = iωµ 0µ H, H = 0, H = iωǫ 0 ǫẽ, & Ẽr) = E 0 e ik r, Hr) = H0 e ik r 8
Plane harmonic electromagnetic waves E = 0, E = µ 0 µḣ, H = 0, H = ǫ0ǫė, & Er,t) = E 0 e ik r ωt), Hr,t) = H 0 e ik r ωt) skip1/2, Re, c.c.) or Ẽ = 0, Ẽ = iωµ 0µ H, H = 0, H = iωǫ 0 ǫẽ, & Ẽr) = E 0 e ik r, Hr) = H0 e ik r exercise) k E 0 = 0, k E 0 = ωµ 0 µh 0, k H 0 = 0, k H 0 = ωǫ 0 ǫe 0, E 0 k, H 0 k, E 0 H 0, transverse waves, ωǫ 0 ǫe 0 = H 0 k orthogonal right-hand system{e 0,H 0,k}. 8
Plane harmonic electromagnetic waves Er,t) = E 0 e ik r ωt), Hr,t) = H 0 e ik r ωt). Example: k = ke z, E 0 = E 0 e x, H 0 = 1 ωµ 0 µ k E 0 = ǫ0 ǫ µ 0 µ E 0e y E 0 R) 9
Plane harmonic electromagnetic waves Er,t) = E 0 e ik r ωt), Hr,t) = H 0 e ik r ωt). Example: k = ke z, E 0 = E 0 e x, H 0 = 1 ωµ 0 µ k E ǫ0 ǫ 0 = µ 0 µ E 0e y E 0 R) ǫ0 ǫ Er,t) = E 0 e x coskz ωt), Hr,t) = µ 0 µ E 0e y coskz ωt). 9
Plane harmonic electromagnetic waves Er,t) = E 0 e ik r ωt), Hr,t) = H 0 e ik r ωt). Example: k = ke z, E 0 = E 0 e x, H 0 = 1 ωµ 0 µ k E ǫ0 ǫ 0 = µ 0 µ E 0e y E 0 R) ǫ0 ǫ Er,t) = E 0 e x coskz ωt), Hr,t) = µ 0 µ E 0e y coskz ωt). ˆǫ = ǫr)1, ˆµ = µr)1!) Griffiths, Fig. 9.10, page 379 B = µ 0 µh, Br,t) = E 0 c m e y coskz ωt). 9
Polarization Er,t) = E 0 e ik r ωt), Hr,t) = H 0 e ik r ωt). Polarization orientation of the electric part of the wave. Complex notation: E 0,H 0 C 3 in general. k,e 0 given,k E 0 H 0 = ωµ 0 µk E 0. 10
Polarization Er,t) = E 0 e ik r ωt), Hr,t) = H 0 e ik r ωt). Polarization orientation of the electric part of the wave. Complex notation: E 0,H 0 C 3 in general. k,e 0 given,k E 0 H 0 = ωµ 0 µk E 0. Assume k = ke z, E 0,x = E 0,x, E 0,y = E 0,y e iδ common phase ine 0,x ande 0,y omitted) Er,t) = E 0,x e x coskz ωt)+ E 0,y e y coskz ωt+δ) 10
Polarization k = ke z, E 0,x = E 0,x, E 0,y = E 0,y e iδ Er,t) = E 0,x e x coskz ωt)+ E 0,y e y coskz ωt+δ) ) 11
Polarization k = ke z, E 0,x = E 0,x, E 0,y = E 0,y e iδ Er,t) = E 0,x e x coskz ωt)+ E 0,y e y coskz ωt+δ) ) δ = 0, δ = ±π: Er,t) = E 0,x e x + E 0,y e y ) coskz ωt), linear polarized wave, E oscillates in the plane { E 0,x e x + E 0,y e y, k} 11
Polarization k = ke z, E 0,x = E 0,x, E 0,y = E 0,y e iδ Er,t) = E 0,x e x coskz ωt)+ E 0,y e y coskz ωt+δ) ) δ = 0, δ = ±π: Er,t) = E 0,x e x + E 0,y e y ) coskz ωt), linear polarized wave, E oscillates in the plane { E 0,x e x + E 0,y e y, k} δ = ±π/2, E 0,x = E 0,y = E 0 : Er,t) = E 0 coskz ωt)e x ±sinkz ωt)e y ) circularly polarized wave, z fixed: Et) describes a circle in time. 11
Polarization k = ke z, E 0,x = E 0,x, E 0,y = E 0,y e iδ Er,t) = E 0,x e x coskz ωt)+ E 0,y e y coskz ωt+δ) ) δ = 0, δ = ±π: Er,t) = E 0,x e x + E 0,y e y ) coskz ωt), linear polarized wave, E oscillates in the plane { E 0,x e x + E 0,y e y, k} δ = ±π/2, E 0,x = E 0,y = E 0 : Er,t) = E 0 coskz ωt)e x ±sinkz ωt)e y ) circularly polarized wave, z fixed: Et) describes a circle in time. otherwise: elliptical polarization, 11
Polarization k = ke z, E 0,x = E 0,x, E 0,y = E 0,y e iδ Er,t) = E 0,x e x coskz ωt)+ E 0,y e y coskz ωt+δ) ) δ = 0, δ = ±π: Er,t) = E 0,x e x + E 0,y e y ) coskz ωt), linear polarized wave, E oscillates in the plane { E 0,x e x + E 0,y e y, k} δ = ±π/2, E 0,x = E 0,y = E 0 : Er,t) = E 0 coskz ωt)e x ±sinkz ωt)e y ) circularly polarized wave, z fixed: Et) describes a circle in time. otherwise: elliptical polarization, ) is the sum of two linearly polarized waves. 11
Energy transport u = 1 E D +H B), 2 S = E H, Er,t) = ReẼr) e iωt D,B, H analogously) S, u oscillate in time. 12
Energy transport u = 1 E D +H B), 2 S = E H, Er,t) = ReẼr) e iωt D,B, H analogously) S, u oscillate in time. t+t 1 Consider time-averaged quantities: ft) = ft ) dt T t u = 1 ) Ẽ 4 Re D + H B, S = 1 ) Ẽ 2 Re H. exercise) 12
Energy transport u = 1 E D +H B), 2 S = E H, Er,t) = ReẼr) e iωt D,B, H analogously) S, u oscillate in time. t+t 1 Consider time-averaged quantities: ft) = ft ) dt T t u = 1 ) Ẽ 4 Re D + H B, S = 1 ) Ẽ 2 Re H. exercise) Specifically: exercise) Er,t) = E 0 e ik r ωt), Hr,t) = H 0 e ik r ωt) u = 1 2 ǫ 0ǫ E 0 2 = 1 2 µ 0µ H 0 2, S = 1 ǫ0 ǫ 2 µ 0 µ E 0 2 k k, S = c mu k k, Intensity S : average power per unit area transported by the wave. 12
Spherical waves 1 2 ) t 2 ψr,t) = 0. 13
Spherical waves 1 2 ) t 2 ψr,t) = 0. ) Spherical coordinatesr,ϑ,ϕ: = 1 r 2 r 2 ) + 1 { 1 r r r 2 sinϑ ϑ sinϑ ) ϑ + 1 sin 2 ϑ 2 } ϕ 2. 13
Spherical waves 1 2 ) t 2 ψr,t) = 0. ) Spherical coordinatesr,ϑ,ϕ: = 1 r 2 r 2 ) + 1 { 1 r r r 2 sinϑ ϑ sinϑ ) ϑ + 1 sin 2 ϑ 2 } ϕ 2. exercise) Spherically symmetric solutions of ): ψr,t) = 1 r f outkr ωt)+ 1 r f inkr+ωt) with ω 2 = k 2 f out, f in arbitrary smooth), ω > 0). 13
Spherical waves ψr,t) = 1 r f outkr ωt)+ 1 r f inkr+ωt). 14
Spherical waves ψr,t) = 1 r f outkr ωt)+ 1 r f inkr+ωt). Wave fronts: spheres around the origin. 14
Spherical waves ψr,t) = 1 r f outkr ωt)+ 1 r f inkr+ωt). Wave fronts: spheres around the origin. Features off out move with velocity c m = ω/k outward, damped 1 r. 14
Spherical waves ψr,t) = 1 r f outkr ωt)+ 1 r f inkr+ωt). Wave fronts: spheres around the origin. Features off out move with velocity c m = ω/k outward, damped 1 r. Features off in move with velocity c m = ω/k inward, growing 1 r. 14
Spherical waves ψr,t) = 1 r f outkr ωt)+ 1 r f inkr+ωt). Wave fronts: spheres around the origin. Features off out move with velocity c m = ω/k outward, damped 1 r. Features off in move with velocity c m = ω/k inward, growing 1 r. Harmonic spherical electromagnetic waves, Er,t) = E 0 r) 1 r eikr ωt), Hr,t) = H 0 r) 1 r eikr ωt),... as for the plane waves: period, wavenumber, wavelength, phase velocity, transverse waves, {E 0 r),h 0 r),r} form a right-hand system, polarization. 14
General solution of the wave equation 1 2 ) t 2 ψr,t) = 0, ψr,0) = ψ 0 r), t ψr,0) = φ 0 r), 15
General solution of the wave equation 1 & ψr,t) = 1 2π) 2 2 ) t 2 ψr,t) = 0, ψr,0) = ψ 0 r), t ψr,0) = φ 0 r), ψk,ω) e ik r ωt) dω d 3 k, 15
General solution of the wave equation 1 2 ) t 2 ψr,t) = 0, ψr,0) = ψ 0 r), t ψr,0) = φ 0 r), & ψr,t) = 1 2π) 2 ψk,ω) e ik r ωt) dω d 3 k, k 2 + ω2 ) ψk,ω) = 0, 15
General solution of the wave equation 1 2 ) t 2 ψr,t) = 0, ψr,0) = ψ 0 r), t ψr,0) = φ 0 r), & ψr,t) = 1 2π) 2 ψk,ω) e ik r ωt) dω d 3 k, k 2 + ω2 ) ψk,ω) = 0, ψk,ω) = a f k)δω ω k )+a b k)δω +ω k ), ω k = c m k, 15
General solution of the wave equation 1 2 ) t 2 ψr,t) = 0, ψr,0) = ψ 0 r), t ψr,0) = φ 0 r), & ψr,t) = 1 2π) 2 ψk,ω) e ik r ωt) dω d 3 k, k 2 + ω2 ) ψk,ω) = 0, ψk,ω) = a f k)δω ω k )+a b k)δω +ω k ), ω k = c m k, ψr,t) = 1 2π) 2 a f k) e ik r ω kt) +ab k) e ik r +ω kt) ) d 3 k, 15
General solution of the wave equation 1 2 ) t 2 ψr,t) = 0, ψr,0) = ψ 0 r), t ψr,0) = φ 0 r), & ψr,t) = 1 2π) 2 ψk,ω) e ik r ωt) dω d 3 k, k 2 + ω2 ) ψk,ω) = 0, ψk,ω) = a f k)δω ω k )+a b k)δω +ω k ), ω k = c m k, ψr,t) = 1 2π) 2 a f k) e ik r ω kt) +ab k) e ik r +ω kt) ) d 3 k, ψr,0) = ψ 0 r), t ψr,0) = φ 0 r)... a f k), a b k). 15
Upcoming Next lecture: Reflection and transmission at interfaces, lossy materials, wave packets, phase and group velocity, dispersion 16