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 ( 29 = ε 0 d v r 1 dt 1 d v r 2 2 χ αβγ (2) v r - v r 1 , t - t 1 ; v r - v r 2 , t - t 2 ( 29 E β v r 1 , t 1 ( 29 E γ v r 2 , t 2 ( 29 t 2 t 1 v r 2 v 1 βγ 0 d v r 1 1 d v r 2 2 d v r 3 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 ( 29 t 3 t 2 t 1 v 3 v 2 v 1 βγδ × E β v r 1 , t 1 ( 29 E γ v r 2 , t 2 ( 29 E δ v r 3 , t 3 ( 29 + L . [ VII-1 ] The wave vector and frequency dependent second and third order susceptibilities are then defined as χ (2) v k 1 , ϖ 1 ; v 2 , ϖ 2 = d v R 1 d τ 1 d v 2 d τ 2 exp - i v 1 v 1 [ ] exp + i ϖ 1 τ 1 [ ] τ 2 τ 1 v 2 v 1 × exp - i v 2 v 2 exp + i ϖ 2 τ 2 [ ] χ v 1 , τ 1 ; v 2 , τ 2 [ VII-2a ] and χ (3) v 1 , ϖ 1 ; v 2 , ϖ 2 ; v 3 , ϖ 3 = d v 1 d τ 1 d v 2 d τ 2 d v 3 d τ 3 exp - i v 1 v 1 exp + i ϖ 1 τ 1 [ ] τ 3 τ 2 τ 1 v 3 v 2 v 1 × exp - i v 2 v 2 exp + i ϖ 2 τ 2 [ ] exp - i v 3 v 3 exp + i ϖ 3 τ 3 [ ] × χ v 1 , τ 1 ; v 2 , τ 2 ; v 3 , τ 3 . [ 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 ( 29 = ε 0 d v k 1 d ϖ 1 d v 2 d ϖ 2 exp i v 1 + v 2 ( 29 v r [ ] exp - i ϖ 1 2 ( 29 t [ ] ϖ 2 ϖ 1 v 2 v 1 βγ ×χ αβγ (2) v 1 , ϖ 1 ; v 2 , ϖ 2 E β v 1 , ϖ 1 E γ v 2 , ϖ 2 0 d v 1 d ϖ 1 d v 2 d ϖ 2 d v 3 d ϖ 3 exp i v 1 + v 2 + v 3 v r [ ] exp - i ϖ 1 2 3 ( 29 t [ ] ϖ 3 ϖ 2 ϖ 1 v 3 v 2 v 1 βγδ × χ αβγδ (3) v 1 , ϖ 1 ; v 2 , ϖ 2 ; v 3 , ϖ 3 E β v 1 , ϖ E γ v 2 , ϖ 2 E δ v 3 , ϖ 3 + 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 We assume that the potential energy of a one-dimensional nonlinear (anharmonic) oscillator may be written V x ( 29 = V 2 x ( 29 + V 3 x ( 29 + V 4 x ( 29 + L = 1 2 M ϖ o 2 x 2 + 1 3 Ma x 3 + 1 4 Mb x 4 + L [ VII-4 ] (See figures on next page) Thus, the equation of motion of a particle moving in that potential becomes ˙ ˙ x ˙ x o 2 x + a x 2 + b x 3 + L = q M E t [ VII-5 ] 26 χ αβγ (2) must vanish for any material that is invariant under inversion, since both r P and r E are vectors, and are thus odd under inversion symmetry. Note also that χ (3) for a given material has the same transformation properties the elastic constants of that material.
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This note was uploaded on 01/31/2011 for the course PHYSICS 108 taught by Professor Staff during the Winter '08 term at UC Davis.

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ls1_unit_7 - ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 58...

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