Lecture 2 - Dispersion in Materials

# Lecture 2 - Dispersion in Materials - Lecture 2 Dispersion...

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Unformatted text preview: Lecture 2: Dispersion in Materials 5 nm Course Info Course webpage • Is now up and running www.ecepurdue.edu/~shalaev Let me know what you think! • Topics ? • Format ? • Direct questions • Ask questions any time………. • Big comments on the nano course are most welcome! Speed of the EM wave: Compare 2 2 1 1 r r c v μ ε ε ε = = Previous Lecture: Maxwell + Wave Equation 2 2 2 r t μ ε ε ∂ ∇ = ∂ E E and general wave Equation: Where c 2 = 1/( ε μ ) = 1/((8.85x10-12 C 2 /m 3 kg) (4 π x 10-7 m kg/C 2 )) = ( 3.0 x 10 8 m/s) 2 ( 29 ( 29 2 2 2 2 , 1 , r t r t v ∂ ∇ = ∂ U U t Refractive index is defined by: Optical refractive index 1 r c n v ε χ = = = + Note: Including polarization results in same wave equation with a different ε r c becomes v 3.0 3.4 index: n Refractive Index Various Materials 2.0 1.0 0.1 1.0 10 λ ( μ m ) Refractive in ( 29 ( 29 ( 29 { } , Re , exp z t z ikz i t ϖ ϖ =- + E E ( 29 ( 29 2 2 2 2 2 , , t n t c ∂ ∇ = ∂ E r E r t Derived from wave equation Dispersion Relations Dispersion relation: ϖ = ϖ (k) Substitute: Result: ϖ 2 2 2 2 n k c ϖ = Check this! 2 2 2 2 c k n ϖ = 1 r g d c c c dk k n v ϖ ϖ ε χ ≡ = = = = + k g v Phase velocity: Group velocity: ph v k ϖ ≡ A z Absorption and Dispersion of EM Waves EM wave: ( 29 ( 29 ( 29 { } , Re , exp z t z ikz i t ϖ ϖ =- + E E 2 2 2 2 c k n ϖ = Dispersion relation k n c ϖ = ± ' '' n n in = + Absorbing materials can be described by a complex n: '' ' '' n in n i n i ϖ ϖ ϖ α ± + = ± + ≡ ±- follows that: Transparent materials can be described by a purely real refractive index n ( 29 ' '' ' '' 2 k n in n i n i c c c β = ± + = ± + ≡ ±- It follows that: Investigate + sign: ( 29 ( 29 , Re , exp 2 z t z i z z i t α ϖ β ϖ =-- + E E Traveling wave Decay ' ' n kn c ϖ β = = Note: n’ acts as a regular refractive index 2 '' 2 '' n kn c ϖ α = - = - α is the absorption coefficient Summary and Future Directions Maxwell’s Equations f ρ ∇ ⋅ = D ∇⋅ = B ∂ ∇× = - ∂ B E t ∂ ∇ × = + ∂ D H J t Curl Equations lead to 2 2 2 2 2...
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Lecture 2 - Dispersion in Materials - Lecture 2 Dispersion...

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