Handout 30

Handout 30 - Handout 30 Optical Processes in Solids and the...

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1 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Handout 30 Optical Processes in Solids and the Dielectric Constant In this lecture you will learn: • Linear response functions • Kramers-Kronig relations • Dielectric constant of solids • Interband and Intraband contributions to the dielectric constant of solids ECE 407 – Spring 2009 – Farhan Rana – Cornell University Dielectric Constant of Solids +ve nucleus + -ve electron cloud (valence band and core electrons) Dielectric in an E-field Dielectric (or Intrinsic Semiconductor) E + ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + Material gets polarized (i.e. develops electric dipoles) E P E D o r r r r ε = + = E P e o r r χ = ( ) e o + = 1
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2 ECE 407 – Spring 2009 – Farhan Rana – Cornell University High Frequency Dielectric Constant of Solids Consider a sinusoidal E&M wave of frequency ω propagating in a solid: () ( ) ( ) { } t i o e r E t r q E n t r E = = , Re . cos ˆ , r r r r r r E r H r q r Where the electric field “phasor” is: () r q i o e E n r E r r r r . ˆ , = Similarly, the magnetic field phasor is: And the two field are related by the two Maxwell equations: ( ) r q i o e H n q r H r r r r . ˆ ˆ , × = ( ) µ , , r H i r E o r r r r = × ( ) ( ) ε , , r E i r H r r r r = × Faraday’s Law Ampere’s Law These two equations together give the dispersion relation of the E&M wave: o o c q q = = r ECE 407 – Spring 2009 – Farhan Rana – Cornell University Refractive Index of Solids The refractive index of a material is defined as: ( ) o n = The wave dispersion relation is then: n c q = And the electric field phasor can be written as: ( ) r q c n i o r q i o e E n e E n r E r r r r r . ˆ . ˆ ˆ , = = The refractive index usually has real and imaginary parts: " ' n i n n + = The electric field phasor is then: ( ) ( ) r q c n r q c n i o e e E n r E r r r r . ˆ ' ' . ˆ ' ˆ , = The imaginary part of the index describes wave decay (or wave amplification if gain is present)
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3 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Imaginary Part of the Refractive Index and the Loss Coefficient Stimulated absorption k r ω h E We have already seen that stimulated absorption results in a wave to decay in a medium (optical loss): () ( ) P R R = α h ( ) [] × = FBZ 3 3 2 2 2 2 ˆ . ' δ π ε h r r r r r r k E k E k E f k E f k d n P c n m e v c c v cv o ( ) r q e r E r r r . ˆ 2 , Where: But we also have: ( ) r q c n e r E r r r . ˆ ' ' , This means the imaginary part of the refractive index is: ( ) 2 ' ' c n = ECE 407 – Spring 2009 – Farhan Rana – Cornell University High Frequency Dielectric Constant of Solids
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This note was uploaded on 10/21/2009 for the course ECE 4070 taught by Professor Rana during the Spring '08 term at Cornell.

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Handout 30 - Handout 30 Optical Processes in Solids and the...

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