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At very high frequencies h binding energy of

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Unformatted text preview: time t=0, then the polarization at any other time is: ∞ Px ( t ) = ∫−∞ χ (ω )e− iωt dω . Since P is real, we may write this as: € ∞ Px ( t ) = ∫ [Re χ (ω ) cosωt + Im χ (ω ) sin ωt ]dω . −∞ This has to be zero for t<0. Therefore, € ∞ ∞ ∫ Re χ (ω ) cosωtdω = − ∫ Im χ (ω ' ) sinω ' tdω ' for t<0. −∞ −∞ For t>0, we may write the same equation but with −t: € ∞ ∞ ∫ Re χ (ω ) cosωtdω = ∫ Im χ (ω ' ) sinω ' tdω ' for t>0. In either case, € € −∞ ∫ −∞ ∞ −∞ ∞ Re χ (ω ) cos ωtdω = sgn t ∫ Im χ (ω ' ) sin ω ' tdω ' . −∞ 6 Now we note that since P and E are real, the susceptibility function satisfies χ(−ω)=χ*(ω), so Re χ is even and Im χ is odd. Therefore, the left ­hand side is the Fourier transform of Re χ. We may thus use the inverse transform, ∞ 1∞ Re χ (ω ) = ∫−∞ sgn t ∫−∞ Im χ (ω ' ) sinω ' tdω ' cosωtdt , 2π To go further, we introduce σ0 as an infintesimal parameter, € ∞ 1 ∞ −σ | t | Re χ (ω ) = ∫−∞ e sgn t ∫−∞ Im χ (ω ' ) sinω ' tdω ' cosωtdt , 2π so that we are justified in performing the integral over t: € ∞ ω ' +ω ω ' −ω ∫−∞ e−σ | t | sgn t sinω ' t cosωtdt = (ω ' +ω )2 + σ 2 + (ω ' −ω )2 + σ 2 . This gives (using the oddness of Im χ to combine the two terms) € 1∞ ω ' −ω Re χ (ω ) = ∫−∞ Im χ (ω ' ) dω ' . π (ω ' −ω ) 2 + σ 2 If we take the limit as σ0, the integral becomes a principal part: € ∞ Im χ (ω ' ) 1 Re χ (ω ) = PP ∫−∞ dω ' , π ω ' −ω where PP indicates that one is to exclude a region |ω’ ­ω|<ς from the integration region, and take the limit as ς0. An analogous relation (swapping Re with Im and € introducing a − sign) allows one to go from Re χ to Im χ. It is possible to express the same equation in terms of only the positive frequencies using the oddness of Im χ: ∞ ω ' Im χ (ω ' ) 2 Re χ (ω ) = PP ∫ 0 dω ' π ω '2 −ω 2 . ∞ Re χ (ω ' ) 2ω Im χ (ω ) = − PP ∫ dω ' 0 ω ' 2 −ω 2 π These are called the Kramers Kronig relations. With the help of the Kramers ­Kronig relations, we need only specify Im χ(ω). € An interesting law can be obtained for the behavior of Im...
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