Lecture course "Integrated Nanophotonics" by Dmitry Fedyanin

Lecture 4: Guiding Light (Part 1)

Transcript

1. Integrated Nanophotonics Dmitry Yu. Fedyanin Laboratory of Nanooptics and Plasmonics, MIPT Lecture 4: Guiding Light (Part 1)
2. 2 BASIC EQUATIONS rot E=i ω c μ H rot H=−i ω c ε E Monochromatic field Plane wave E x =A ei K z H y = 1 ζ A ei K z
3. 3 BASIC EQUATIONS rot E=i ω c μ H rot H=−i ω c ε E Monochromatic field Plane wave E x =A ei K z H y = 1 ζ A ei K z Δ E+K2 E=0,div E=0 Δ H+K2 H=0,div H=0 F=A ei K (α x+β y+γ z) α2 +β2 +γ2 =1 Generalized plane wave F=A ei K r r=α x+β y+γ z
4. 4 AT THE INTERFACE F=A 1 ei K 1 (α1 x+β1 y+γ1 z) α1 2 +β1 2 +γ1 2 =1 Umov-Poynting equation dW dt +P+Σ=0 F 2 =A 2 ei K 2 (α2 x+β2 y+γ2 z) α2 2 +β2 2 +γ2 2 =1
5. 5 AT THE INTERFACE F=A 1 ei K 1 (α1 x+β1 y+γ1 z) α1 2 +β1 2 +γ1 2 =1 F 2 =A 2 ei K 2 (α2 x+β2 y+γ2 z) α2 2 +β2 2 +γ2 2 =1 - J. Zenneck, Über die Fortpflanzung ebener elektromagnetischer Wellen längs einer ebenen Leiterfläche und ihre Beziehung zur drahtlosen Telegraphie // Ann. der Physik, 23, 846–866, (1907).
6. 6 AT THE INTERFACE Two polarizations: TM and TE At the interface: E x | z=−0 =E x | z=+0 ε1 E z | z=−0 =ε2 E z | z=+0 H y | z=−0 =H y | z=+0 H x | z=−0 =H x | z=+0 H z | z=−0 =H z | z=+0 E y | z=−0 =E y | z=+0
7. 7 SURFACE WAVE TM wave E x | z=−0 =E x | z=+0 ε1 E z | z=−0 =ε2 E z | z=+0 H y | z=−0 =H y | z=+0 E x =A 1 ei K 1 (α1 x+γ1 z) , z>0 E x =A 2 ei K 2 (α2 x−γ2 z) , z<0 K 1 α1 =K 2 α2 A 1 =A 2 H y | z=−0 =H y | z=+0
8. 8 SURFACE WAVE TM wave E x | z=−0 =E x | z=+0 ε1 E z | z=−0 =ε2 E z | z=+0 H y | z=−0 =H y | z=+0 E x =A 1 ei K 1 (α1 x+γ1 z) , z>0 E x =A 2 ei K 2 (α2 x−γ2 z) , z<0 H y =−i ω c ε1 κ1 A 1 ei K 1 α1 x e−κ1 z ,z>0; H y =i ω c ε2 κ2 A 2 ei K 2 α2 x eκ2 z , z<0 α2 +γ2 =1 κ=√(K α)2 −K2 K 1 α1 =K 2 α2 A 1 =A 2 H y | z=−0 =H y | z=+0 E x =A 1 ei K 1 x e−κ1 z , z>0; E x =A 2 ei K 2 x eκ2 z ,z<0
9. 9 SURFACE WAVE TM wave E x | z=−0 =E x | z=+0 ε1 E z | z=−0 =ε2 E z | z=+0 H y | z=−0 =H y | z=+0 E x =A 1 ei K 1 (α1 x+γ1 z) , z>0 E x =A 2 ei K 2 (α2 x−γ2 z) , z<0 H y =−i ω c ε1 κ1 A 1 ei K 1 α1 x e−κ1 z ,z>0; H y =i ω c ε2 κ2 A 2 ei K 2 α2 x eκ2 z , z<0 α2 +γ2 =1 κ=√(K α)2 −K2 K 1 α1 =K 2 α2 A 1 =A 2 H y | z=−0 =H y | z=+0 E x =A 1 ei K 1 x e−κ1 z , z>0; E x =A 2 ei K 2 x eκ2 z ,z<0
10. 10 SURFACE WAVE TM wave β=ω c √ ε1 ε2 ε1 +ε2 α2 +γ2 =1 K=ω c β=K α κ= √β2 −(ω c )2 ε − ε1 κ1 = ε2 κ2
11. 11 SURFACE WAVE TE wave does not exist
12. 12 SURFACE WAVE TM wave β=ω c √ ε1 ε2 ε1 +ε2 α2 +γ2 =1 K=ω c β=K α κ= √β2 −(ω c )2 ε − ε1 κ1 = ε2 κ2 ε1 =ε0 + 4 πσ ω i, 4πσ ω ≫1 ε2 =εd
13. 13 SURFACE WAVE TM wave β=ω c √ ε1 ε2 ε1 +ε2 α2 +γ2 =1 K=ω c β=K α κ= √β2 −(ω c )2 ε − ε1 κ1 = ε2 κ2 ε1 =ε0 + 4 πσ ω i, 4 π σ ω ≫1 ε2 =εd β≈ω c √εd √1+ εd 4 πσ ω i≈ ω c √εd (1+ εd 8 πσ ω i) 1 κ2 ≈ 1 ω c √εd √4 πσ ω εd 1−i √2 ; 1 κ1 ≈ 1 ω c √4 πσ ω 1+i √2
14. 14 SURFACE WAVE TM wave β=ω c √ ε1 ε2 ε1 +ε2 − ε1 κ1 = ε2 κ2 ε1 =εr − ωp 2 ω2 +i γω ε2 =εd
15. 15 SURFACE WAVE TM wave β=ω c √ ε1 ε2 ε1 +ε2 − ε1 κ1 = ε2 κ2 ε1 =εm =εr − ωp 2 ω2 +i γω ε2 =εd 1 κ2 ≈ 1 ω c √εd √ωp 2 ω2 −εd −εr εd = λd 2π √|εm| εd −1≈ λd 2π √|εm| εd 1 κ1 = 1 ω c √− 1 ε1 − ε2 ε1 2 ≈ c ωp
16. 16 SURFACE WAVE TM wave β=ω c √ ε1 ε2 ε1 +ε2 − ε1 κ1 = ε2 κ2 ε1 =εr − ωp 2 ω2 +i γω ε2 =εd γ=0
17. 17 THREE-LAYER STRUCTURE TM and TE waves
18. 18 THREE-LAYER STRUCTURE TM and TE waves we will derive the dispersion relation only for the TM wave z>a {H y =A eik x x e−κ3 z E x =− κ3 iε3 k A eik x x e−κ3 z z<−a {H y =Bei k x x eκ2 z E x = κ2 iε2 k B ei k x x eκ2 z −a<z<a {H y =C eik x x eκ1 z +Dei k x x e−κ1 z E x = κ1 iε1 k (C ei k x x eκ1 z −D eik x x e−κ1 z)
19. 19 THREE-LAYER STRUCTURE Boundary condition at z = −a {B e−κ2 a =C e−κ1 a +Deκ1 a κ2 ε2 Be−κ2 a = κ1 ε1 (C e−κ1 a −D eκ1 a) Boundary condition at z = a {A e−κ3 a =C eκ1 a +D e−κ1 a κ3 ε3 A e−κ3 a = κ1 ε1 (−C eκ1 a +De−κ1 a)
20. 20 THREE-LAYER STRUCTURE Boundary condition at z = −a {B e−κ2 a =C e−κ1 a +Deκ1 a κ2 ε2 Be−κ2 a = κ1 ε1 (C e−κ1 a −D eκ1 a) Boundary condition at z = a {A e−κ3 a =C eκ1 a +D e−κ1 a κ3 ε3 A e−κ3 a = κ1 ε1 (−C eκ1 a +De−κ1 a) Dispersion relation
21. 21 THREE-LAYER STRUCTURE TM wave TE wave e−4κ1 a = κ1 +κ2 κ1 −κ2 κ1 +κ3 κ1 −κ3
22. 22 THREE-LAYER STRUCTURE ε2 =ε3 Symmetric structure
23. 23 THREE-LAYER STRUCTURE TM wave TE wave ε2 =ε3 Symmetric structure tanh(κ1 a)=− κ2 ε1 κ1 ε2 tanh(κ1 a)=− κ1 ε2 κ2 ε1 tanh(κ1 a)=− κ2 κ1 tanh(κ1 a)=− κ1 κ2
24. 24 THREE-LAYER STRUCTURE Different approach wikipedia β=k n 1 sin θ, sinθ > n 2 n 1 , n 3 n 1
25. 25 THREE-LAYER STRUCTURE Different approach β=k n 1 sin θ, sinθ > n 2 n 1 , n 3 n 1 Phase shift due to reflection at the interfaces (Goos-Hanchen shift) Dispersion relation 4 k n 1 acosθ −ϕ12 −ϕ13 =2π N , N=0,1,2,3,...
26. 26 NEXT CLASS Light propagation in photonic and nanophotonic waveguides Homework: Л. А. Вайнштейн, Электромагнитные волны. — М.: Радио и связь, 1988. Chapters 1, 2, 4: Paragraphs 8, 11-16, 24, 25