Lecture course "Integrated Nanophotonics" by Dmitry Fedyanin

Lecture 3: Fundamentals of Light Propagation

Transcript

1. Integrated Nanophotonics Dmitry Yu. Fedyanin Laboratory of Nanooptics and Plasmonics, MIPT Lecture 3: Fundamentals of Light Propagation
2. 2 BASIC EQUATIONS ∂ρ ∂t +div j=0 continuity equation
3. 3 BASIC EQUATIONS ∂ρ ∂t +div j=0 continuity equation div D=4 πρ Poisson equation
4. 4 BASIC EQUATIONS ∂ρ ∂t +div j=0 continuity equation div D=4 πρ Poisson equation div B=0 Gauss law for magnetic field
5. 5 BASIC EQUATIONS ∂ρ ∂t +div j=0 Continuity equation div D=4 πρ Poisson equation div B=0 Gauss law for magnetic field rot E=− 1 c ∂ B ∂t Maxwell–Faraday equation
6. 6 BASIC EQUATIONS ∂ρ ∂t +div j=0 Continuity equation div D=4 πρ Poisson equation div B=0 Gauss law for magnetic field rot E=− 1 c ∂ B ∂t Maxwell–Faraday equation rot H= 1 c ∂ D ∂t + 4 π c j Ampere circuital law with Maxwell's addition These equations are valid for all kinds of media!
7. 7 BASIC EQUATIONS D=εE B=μ H j=σ E These equations are not universal and may be incorrect in some media! So-called material equations D 2 n−D 1 n=4 πρsurf [n E 2 ]−[nE 1 ]=0 B 2 n−B 1 n=0 [n H 2 ]−[n H 1 ]= 4 π c j surf Boundary conditions
8. 8 ENERGY OF THE ELECTROMAGNETIC FIELD W = 1 8π ∫ V (εE2 +μ H2)dV P=∫ V j EdV Σ= c 4 π ∮ S [EH ]ndS Energy Absorption/Generation Energy flux
9. 9 ENERGY OF THE ELECTROMAGNETIC FIELD W = 1 8π ∫ V (εE2 +μ H2)dV P=∫ V j EdV Σ= c 4 π ∮ S [EH ]ndS Energy Absorption/Generation Energy flux dW dt +P+Σ=0 Umov-Poynting equation The energy conservation law in electrodynamics Umov-Poynting equation
10. 10 ENERGY OF THE ELECTROMAGNETIC FIELD W =∫ V w dV =∫ V (w e +w m )dV P=∫ V pdV Σ=∮ S SndS Energy Absorption/Generation Energy flux dw dt +p+div S=0 Umov-Poynting equation The energy conservation law in electrodynamics Umov-Poynting equation
11. 11 MONOCHROMATIC FIELD div D=4 πρ rot E=− 1 c ∂ B ∂t div B=0 rot H= 1 c ∂ D ∂t + 4 π c j E(t)=Ecos(ωt)
12. 12 MONOCHROMATIC FIELD div D=4 πρ rot E=− 1 c ∂ B ∂t div B=0 rot H= 1 c ∂ D ∂t + 4 π c j E(t)=Ecos(ωt) E(t)=Ecos(ωt+φ)=Re{Eexp[−i(ωt+φ)]}→ →E(t)=Re{Eexp[−i ωt]} H (t)=Re {H exp[−iωt ]} . .. In photonics, exp(-iωt) is more convenient than exp(iωt).
13. 13 MONOCHROMATIC FIELD div D=4 πρ rot E=− 1 c ∂ B ∂t div B=0 rot H= 1 c ∂ D ∂t + 4 π c j E(t)=Ecos(ωt) E(t)=Ecos(ωt+φ)=Re{Eexp[−i(ωt+φ)]}→ →E(t)=Re{Eexp[−i ωt]} H (t)=Re {H exp[−iωt ]} . .. div Dexp(−iωt)=4πρexp(−i ωt) rot Eexp(−iωt)=i ω c Bexp(−iωt) div Bexp(−iωt)=0 rot H exp(−i ωt)=−i ω c Dexp(−i ωt) + 4 π c j exp(−iωt)
14. 14 MONOCHROMATIC FIELD div Dexp(−iωt)=4πρexp(−i ωt) rot Eexp(−iωt)=i ω c Bexp(−iωt) div Bexp(−iωt)=0 rot H exp(−i ωt)=−i ω c Dexp(−i ωt) + 4 π c j exp(−iωt)
15. 15 MONOCHROMATIC FIELD div Dexp(−iωt)=4πρexp(−i ωt) rot Eexp(−iωt)=i ω c Bexp(−iωt) div Bexp(−iωt)=0 rot H exp(−i ωt)=−i ω c Dexp(−i ωt) + 4 π c j exp(−iωt)
16. 16 MONOCHROMATIC FIELD div Dexp(−iωt)=4πρexp(−i ωt) rot Eexp(−iωt)=i ω c μ H exp(−i ωt) div Bexp(−iωt)=0 rot H exp(−i ωt)=−i ω c εEexp(−i ωt) + 4 π c σ Eexp(−i ωt) D(ω)=ε(ω)E(ω) B(ω)=μ(ω)H(ω) j(ω)=σ(ω)E(ω)
17. 17 MONOCHROMATIC FIELD div Dexp(−iωt)=4πρexp(−i ωt) rot Eexp(−iωt)=i ω c μ H exp(−i ωt) div Bexp(−iωt)=0 rot H exp(−i ωt)=−i ω c εEexp(−i ωt) + 4 π c σ Eexp(−i ωt)
18. 18 MONOCHROMATIC FIELD rot Eexp(−iωt)=i ω c μ H exp(−i ωt) rot H exp(−i ωt)=−i ω c εEexp(−i ωt)+ 4π c σ Eexp(−i ωt)
19. 19 MONOCHROMATIC FIELD rot E=i ω c μ H rot H=−i ω c ε E+ 4 π c σ E
20. 20 COMPLEX DIELECTRIC FUNCTION rot E=i ω c μ H rot H=−i ω c (ε0 + 4 πσ ω i)E ε=ε0 + 4 πσ ω i Complex dielectric function δ is the delay between D and E ε = Reε+i Imε = |ε|ei δ From the macroscopic point of view, there is no difference between the delay and electron conductivity, both lead to heat generation.
21. 21 COMPLEX DIELECTRIC FUNCTION rot E=i ω c μ H rot H=−i ω c (ε0 + 4 πσ ω i)E ε=ε0 + 4 πσ ω i Complex dielectric function δ is the delay between D and E ε = Reε+i Imε = |ε|ei δ From the macroscopic point of view, there is no difference between the delay and electron conductivity, both lead to heat generation. Generally speaking, μ can also be complex, but at optical frequencies μ is usually equal to 1.
22. 22 COMPLEX DIELECTRIC FUNCTION q e = 1 4 π ∮EdD =− 1 4 π |ε| E 0 2∫ 0 2π cos(ωt)sin(ωt−δ)d(ωt) = 1 4 |ε| E 0 2 sin(δ) = 1 4 ε' ' E 0 2 Heat generation E(t)=E 0 cos(ωt) D(t)=|ε|E 0 cos(ωt−δ) q e
23. 23 COMPLEX DIELECTRIC FUNCTION q e = 1 4 π ∮EdD =− 1 4 π |ε| E 0 2∫ 0 2π cos(ωt)sin(ωt−δ)d(ωt) = 1 4 |ε| E 0 2 sin(δ) = 1 4 ε' ' E 0 2 Heat generation p e = q e T = ω 8π ε' ' E 0 2 Heating power E(t)=E 0 cos(ωt) D(t)=|ε|E 0 cos(ωt−δ)
24. 24 COMPLEX DIELECTRIC FUNCTION Dielectric function of the metal at optical frequencies m ¨ x+mγ ˙ x=−q E D=εE D=E+4 π P, P=χ E Derive ε yourself Density of electrons per unit volume is N
25. 25 COMPLEX DIELECTRIC FUNCTION Dielectric function of the metal at optical frequencies m ¨ x+mγ ˙ x=−q E ¨ x+γ ˙ x=− q E m −ω2~ x−i ω γ~ x=− q~ E m E= ~ Ee−i ωt →x=~ x e−i ωt (ω2 +i ω γ)~ x= q~ E m ~ x= q m 1 ω2 +iω γ ~ E ε=1− 4π N q2 m 1 ω2 +i ω γ =1− ωp 2 ω2 +i ω γ → ε=ε∞ − ωp 2 ω2 +i ω γ P=−q x D=εE D=E+4 π P, P=χ E Density of electrons per unit volume is N
26. 26 ELECTROMAGNETIC WAVES Plane waves In a plane wave, complex amplitudes E and H are not functions of x and y. They do depend on z (direction of propagation). rot E=i ω c μ H rot H=−i ω c ε E
27. 27 ELECTROMAGNETIC WAVES Plane waves In a plane wave, complex amplitudes E and H are not functions of x and y. They do depend on z (direction of propagation). rot E=i ω c μ H rot H=−i ω c ε E 0 0
28. 28 ELECTROMAGNETIC WAVES Plane waves In a plane wave, complex amplitudes E and H are not functions of x and y. They do depend on z (direction of propagation). d2 F dz2 +(k √εμ)2 F=0 n=√εμ refractive index n=√|ε||μ|exp(i δ+Δ 2 )
29. 29 ELECTROMAGNETIC WAVES d2 F dz2 +(k √εμ)2 F=0 d2 F dz2 +K2 F=0 ζ = √μ ε wave impedance
30. 30 ELECTROMAGNETIC WAVES Two polarizations where d2 F dz2 +(k √εμ)2 F=0 d2 F dz2 +K2 F=0
31. 31 ELECTROMAGNETIC WAVES where attenuation phase υph = ω K ' = c n' phase velocity L= 1 2 K ' ' propagation length
32. 32 ELECTROMAGNETIC WAVES energy flux S z = c 8π (E x H y * −E y H x *) energy velocity W=W (E, H) energy density υE = Re S z W υph = ω K ' = c n' = c √|ε||μ|cos(i δ+Δ 2 ) phase velocity can be either positive or negative S z = c 8πζ* | A|2 e−2 K ' ' z = c 8π|ζ| cos(δ−Δ 2 )| A|2 e−2K ' ' z Direction of the energy flux coincides with the attenuation direction, i.e. υ E is always positive.
33. 33 REFLECTION AT THE INTERFACE z<0 { E x =A (ei K 1 z +R e−i K 1 z ) H y = 1 ζ1 A (ei K 1 z −R e−i K 1 z ) z>0 { E x =A T ei K 2 z H y = 1 ζ2 A T ei K 2 z
34. 34 REFLECTION AT THE INTERFACE z<0 { E x =A (ei K 1 z +R e−i K 1 z ) H y = 1 ζ1 A (ei K 1 z −R e−i K 1 z ) z>0 { E x =A T ei K 2 z H y = 1 ζ2 A T ei K 2 z E x | z=−0 =E x | z=+0 H y | z=−0 =H y | z=+0 Boundary conditions at the interface T=1+R 1 ζ1 (1−R)= 1 ζ2 (1+R) R= ζ2 −ζ1 ζ2 +ζ1 = ζ2 /ζ1 ζ2 /ζ1 +1 T= 2ζ2 /ζ1 ζ2 /ζ1 +1 Transmittance T= Re E x H y | z=+0 Re E x H y | z=−0 = T2 /ζ2 1/ζ1 = 4 ζ2 /ζ1 (ζ2 /ζ1 +1)2
35. 35 NEXT CLASS Electromagnetic energy and its relation to the dielectric function Group velocity versus energy velocity Light propagation in waveguides Homework: Л. А. Вайнштейн, Электромагнитные волны. — М.: Радио и связь, 1988. Chapter 1: Paragraphs 3, 4, 5, 6, 10