ECE615_Lecture_18 - Lecture 18 Second Harmonic Susceptibility Resonant Case Vij = −(i | er ⋅ E | i = −eE ⋅ r ψ i(r ψ i(r)d 3 r =

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Unformatted text preview: Lecture 18 Second Harmonic Susceptibility: Resonant Case ɶ ɶ Vij = −(i | er ⋅ E | i ) = −eE ⋅ ∫ r ψ i* (r ) ψ i (r )d 3 r = 0 (0) ρ11 = 1 ω 2ω ω iˆ ɺ ˆ ρ nm = −iωnm ρ nm − V , ρ − γ nm ρ nm nm ℏ i ɺ nm = −iωnm ρ nm − ∑ (Vnv ρ vm − ρ nvVvm ) − γ nm ρ nm ρ ℏv i i i i ∂ (1) (0) + iω21 + γ 21 ρ 21 = − V21 ρ11 − V23 ρ31 + ρ 22V21 + ρ 23V31 ℏ ℏ ℏ ℏ ∂t V21 = − µ 21,γ Eγ exp ( −iωt ) + c.c. ∂ (1) i + i (ω21 − ω ) + γ 21 δ 21 = µ 21,γ Eγ ℏ ∂t µ21,γ Eγ 1 (1) ρ 21 = exp [ −iωt ] ℏ −iγ 21 + (ω21 + ω ) (1) ρ 21 = δ 21 exp [ −iωt ] ρ 21 is the only element in the 1st order. 1 ɺ ρ31 = −iω31 ρ31 − i ∑ (V3v ρv1 − ρ3vVv1 ) − γ 31ρ31 ℏv V11 = V33 = V22 = 0 off-resonant higher-order symmetry V33 ρ31 + V32 ρ 21 + V31 ρ11 − ρ33V31 − ρ32V21 − ρ31V11 symmetry i ∂ (2) (1) + iω31 + γ 31 ρ31 = − V32 ρ 21 ℏ ∂t (2) (2) ρ31 = δ 31 exp [ −2iωt ] V32 = − µ32,α Eα exp ( −iωt ) + c.c. i µ32, β Eβ µ 21,γ Eγ (2) γ 31 + i (ω31 − 2ω ) δ 31 = 2 ℏ (ω21 − ω ) − iγ 21 (2) ρ31 = µ32, β µ21,γ Eβ Eγ i exp ( −2iωt ) 2 ℏ (ω21 − ω ) − iγ 21 (ω31 − 2ω ) − iγ 31 ˆˆ µα (t ) = tr ( ρµ ) = ρ31µ13,α + c.c. = dα (ω ) exp ( −2iωt ) + c.c. (2) P = Ndα (ω ) = ε 0 χαβγ ( −2ω ; ω , ω ) Eβ Eγ α ( 2) P = Ndα (ω ) = ε 0 χαβγ ( −2ω ; ω , ω ) Eβ Eγ α µ13,α µ32, β µ21,γ N χαβγ (−2ω; ω , ω ) = 2ε 0 ℏ 2 (ω21 − ω ) − iγ 21 (ω31 − 2ω ) − iγ 31 (2) (0) (1) (2) ρ11 → ρ 21 → ρ31 V21 V32 Non-resonant estimate: χ (2) ∼ 10−14 m / V e = 1.602 × 10−19 C a0 = 5.292 ×10−11 m ℏ = 1.054 ×10−34 J ⋅ s µ ≅ ea0 = 8.478 × 10−30 C ⋅ m ε 0 = 8.85 ×10−12 F/m ω = 1016 s -1 N ≅ 1028 m 3 = 1022 cm3 (2) (3) Explore general cases of χ , χ on your own in text: 3.6, 3.7 ...
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This note was uploaded on 01/10/2011 for the course ECE 615 taught by Professor Shalaev during the Fall '10 term at Purdue University-West Lafayette.

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ECE615_Lecture_18 - Lecture 18 Second Harmonic Susceptibility Resonant Case Vij = −(i | er ⋅ E | i = −eE ⋅ r ψ i(r ψ i(r)d 3 r =

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