ls1_unit_7

ls1_unit_7 - ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 58...

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ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 58 R. Victor Jones, February 22, 2000 VII. NONLINEAR OPTICS -- CLASSICAL PICTURE: A N E XTENDED P HENOMENOLOGICAL MODEL OF POLARIZATION : As an introduction to the subject of nonlinear optical phenomena, we write, in the spirit of Equation [ I-4 ], the most general form of higher order terms in the phenomenological electric field expansion of the polarization density (which may then be inserted in Equations [ I-3 ]) as P α (NL) v r , t ( ) = ε 0 d v r 1 dt 1 d v r 2 dt 2 χ αβγ (2) v r v r 1 , t t 1 ; v r v r 2 , t t 2 ( ) E β v r 1 , t 1 ( ) E γ v r 2 , t 2 ( ) t 2 t 1 v r 2 v r 1 βγ + ε 0 d v r 1 dt 1 d v r 2 dt 2 d v r 3 dt 3 χ αβγδ (3) v r v r 1 , t t 1 ; v r v r 2 , t t 2 ; v r v r 3 , t t 3 ( ) t 3 t 2 t 1 v r 3 v r 2 v r 1 βγδ × E β v r 1 , t 1 ( ) E γ v r 2 , t 2 ( ) E δ v r 3 , t 3 ( ) + L . [ VII-1 ] The wave vector and frequency dependent second and third order susceptibilities are then defined as χ αβγ (2) v k 1 , ω 1 ; v k 2 , ω 2 Λ Ν Μ Ξ Π Ο = d v R 1 d τ 1 d v R 2 d τ 2 exp i v k 1 v R 1 [ ] exp + i ω 1 τ 1 [ ] τ 2 τ 1 v R 2 v R 1 × exp i v k 2 v R 2 [ ] exp + i ω 2 τ 2 [ ] χ αβγ (2) v R 1 , τ 1 ; v R 2 , τ 2 Λ Ν Μ Ξ Π Ο [ VII-2a ] and χ αβγδ (3) v k 1 , ω 1 ; v k 2 , ω 2 ; v k 3 , ω 3 Λ Ν Μ Ξ Π Ο = d v R 1 d τ 1 d v R 2 d τ 2 d v R 3 d τ 3 exp i v k 1 v R 1 [ ] exp + i ω 1 τ 1 [ ] τ 3 τ 2 τ 1 v R 3 v R 2 v R 1 × exp i v k 2 v R 2 [ ] exp + i ω 2 τ 2 [ ] exp i v k 3 v R 3 [ ] exp + i ω 3 τ 3 [ ] × χ αβγδ (3) v R 1 , τ 1 ; v R 2 , τ 2 ; v R 3 , τ 3 Λ Ν Μ Ξ Π Ο . [ VII-2b ]
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ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 59 R. Victor Jones, February 22, 2000 Thus, we may write quite generally 26 P α (NL) v r , t ( ) = ε 0 d v k 1 d ω 1 d v k 2 d ω 2 exp i v k 1 + v k 2 ( ) v r [ ] exp i ω 1 + ω 2 ( ) t [ ] ω 2 ω 1 v k 2 v k 1 βγ ×χ αβγ (2) v k 1 , ω 1 ; v k 2 , ω 2 Λ Ν Μ Ξ Π Ο E β v k 1 , ω 1 Λ Ν Μ Ξ Π Ο E γ v k 2 , ω 2 Λ Ν Μ Ξ Π Ο + ε 0 d v k 1 d ω 1 d v k 2 d ω 2 d v k 3 d ω 3 exp i v k 1 + v k 2 + v k 3 ( ) v r [ ] exp i ω 1 + ω 2 + ω 3 ( ) t [ ] ω 3 ω 2 ω 1 v k 3 v k 2 v k 1 βγδ × χ αβγδ (3) v k 1 , ω 1 ; v k 2 , ω 2 ; v k 3 , ω 3 Λ Ν Μ Ξ Π Ο E β v k 1 , ω Λ Ν Μ Ξ Π Ο E γ v k 2 , ω 2 Λ Ν Μ Ξ Π Ο E δ v k 3 , ω 3 Λ Ν Μ Ξ Π Ο + L . . [ VII-3 ] A S IMPLE CLASSICAL MODEL OF NONLINEAR OPTICAL RESPONSE A simple Lorentz-Dude model is often used in the literature as a valuable guide to the understanding of the frequency behavior of the nonlinear dielectric response. 27
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ls1_unit_7 - ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 58...

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