25. Optical properties of materials-metal

Transcription

1 5. Otical roerties of materials-metal Drue Moel Conuction Current in Metals EM Wave Proagation in Metals Sin Deth Plasma Frequency

2 Drue moel Drue moel : Lorenz moel (Harmonic oscillator moel) without restoration force (that is, free electrons which are not boun to a articular nucleus) Remin! Linear Dielectric Resonse of Matter

3 Conuction Current in Metals The equation of motion of a free electron (not boun to a articular nucleus; C 0), r m e r v 1 14 me Cr ee m ( : relaxation time 10 ) e + meγv ee τ s t τ t t γ Lorentz moel (Harmonic oscillator moel) If C 0 Drue moel (free-electron moel) The current ensity is efine A C J Nev m sim : Substituting in the equation of motion we obtain : J t N e + γ J E me

4 J t N e + γ J E me Assume that the alie electric fiel an the conuction current ensity are given by : E E ex i t J J i t ( ) ex( ) 0 0 Substituting into the equation of motion we obtain : Local aroximation to the current-fiel relation J0 ex( i t) + γ J0ex it ij0ex it + γ J0ex it t Ne E0 ex( i t) me ( ) ( ) ( ) ( + ) Multilying through by ex i t : Ne i + J0 E0, or equivalently me ( γ) Ne i + J E me ( γ)

5 Ne i + J E me ( γ) For static fiels ( 0), Ne Ne J E σ E, where σ : static conuctivity meγ meγ For the general case of an oscillating alie fiel : J σ E 1 σ Ne / me σ E, where σ : ynamic conuctivity γ γ i ( i/ γ) 1 ( i / ) ( γ ) <<, For very low frequencies, 1 the ynamic conuctivity is urely real an the electrons follow the electric fiel. As the frequency of the alie fiel increases, the inertia of electrons introuces a hase lag in the electron resonse to the fiel, an the ynamic conuctivity is comlex. For very high frequencies, 1 J i E e σ E i π ( γ) >>, σ ( ) the ynamic conuctivity is urely imaginary an the electron oscillations are 90 out of hase with the alie fiel.

6 Proagation of EM Waves in Metals Maxwell ' s relations give us the following wave equation for metals : J c t c t P 0, J 0 1 E 1 E + 0 σ But, J E 1 ( i/ γ) Substituting in the wave equation we obtain : σ E c t c i t 1 E 1 E ( / γ) The wave equation is satisfie by electric fiels of the form : E E0 ex i( r t) σ μ0 + i c 1 ( i/ γ), where c 1 μ 0 0

7 Sin Deth at low frequency Consier the case where is small enough that is given by σμ 0 π + i i σμ0 ex i σμ0 c 1 ( i/ γ) 0 Then, π π π π σ μ ex i σμ0 ex i σμ0 cos + i sin σμ0 ( 1+ i) : σμ0 c σc μ0 σ R I nr ni R, I 0 In the metal, for a wave roagating in the z irection : z E E0ex( iz ) E0ex ( Iz) ex i( Rz t) E0ex ex i( Rz t) δ The sin eth δ is given by : 1 δ σμ I 0 0c σ C s For coer the static conuctivity σ Ω m δ 0.66μm J m

8 Refractive Inex of a metal σμ 0 Now consier again the general case, + i c 1 ( i/ γ) c σ c μ 0 iγ σ c μ 0 n 1+ i 1 i + 1 ( i / γ) i γ 1 ( i / γ) γσc μ0 n 1 + iγ The lasma frequency is efine γσ μ0 γ, c Ne c μ0 meγ Ne m e 0 The refractive inex of the conuctive meium is given by, n 1, Ne where + iγ me0 : Plasma frequency

9 What is is the lasma Frequency? If the electrons in a lasma are islace from a uniform bacgroun of ions, electric fiels will be built u in such a irection as to restore the neutrality of the lasma by ulling the electrons bac to their original ositions. Because of their inertia, the electrons will overshoot an oscillate aroun their equilibrium ositions with a characteristic frequency nown as the lasma frequency. E σ / Ne( δ x) / : electrostatic fiel by small charge searation δ x s o o o δ x δ x ex( i t) : small-amlitue oscillation o m ( δ x) Ne Ne ( e) E s m t m o o Ne m o

10 Critical wavelength (or, lasma wavelength) λ λ c πc Born an Wolf, Otics, age 67.

11 Refractive Inex of a metal n 1 + i γ For a high frequency ( >> γ ), n 1 by neglecting γ. < > : n is comlex an raiation is attenuate. EM waves with lower frequencies are reflecte/absorbe at metal surfaces. : n is real an raiation is not attenuate(transarent). EM waves with higher frequencies can roagate through metals.

12 Disersion of Refractive Inex for coer

13 Dielectric constant of metal given by Drue moel ( ) + i n R I + ( nr ini) 1 + ( nr ni ) inrni + iγ γ 1 + i 3 + γ + γ >> γ 1 τ ( ) 1 + i 3 / γ

14 Ieal case : metal as a free-electron gas no ecay (infinite relaxation time) no interban transitions ( ) ( ) 1 τ γ 0 r 1 0

15 An alication of Drue moel : Surface lasmons Plasma wave (oscillation) ensity fluctuation of charge articles Plasmon lasma wave with well efine oscillation frequency (energy) Plasmon in metals collective oscillation of free electrons with well efine energy Surface lasmons Plasmons on metal surfaces

16 Plasma waves (lasmons) Plasma oscillation ensity fluctuation of free electrons Plasmons roagating through bul meia with a resonance at Bul Plasmons rue Ne m Plasmons confine but roagating on surfaces Surface Plasmons (SP) Plasmons confine at nanoarticles with a resonance at Localize Surface Plasmons rue article 1 3 Ne m 0

17 Disersion relation for EM waves in electron gas (bul lasmons) Disersion relation: ( )

18 Disersion relation for surface lasmons m TM wave Z > 0 Z < 0 At the bounary (continuity of the tangential E x, H y, an the normal D z ): E xm Ex Hym Hy mezm Ez

19 Disersion relation for surface lasmons zm z m xm x E E ym y H H xm m ym zm E H y z ym m zm H H ),0, ( zi i xi i E i E i ),0, ( yi xi yi zi H i H i xi i yi zi E H x y z E H

20 Disersion relation for surface lasmons For any EM wave: zi i x + zi x xm x, where c x SP Disersion Relation x c m + m

21 Disersion relation: relation for surface lasmons x-irection: z-irection: m x ' x+ i" x c m + zi i x c 1/ + i ' " m m m i zi ' zi + izi ± c m + 1/ For a boun SP moe: zi must be imaginary: m + < 0 zi ± i x ± x i x > i i c c c + for z < 0 - for z > 0 x must be real: m < 0 So, ' m <

22 Plot of the isersion relation 1 ) ( m m x m c + Plot of the ielectric constants: Plot of the isersion relation: s m + 1,, When x ) (1 ) ( s x c +

23 Surface lasmon isersion relation: x m c m + 1/ zi i c m + 1/ + c x c x Raiative moes (' m > 0) real x real z Quasi-boun moes ( < ' m < 0) imaginary x real z 1+ z Dielectric: x Metal: m ' m + " m Boun moes (' m < ) real x imaginary z λ x ~λ Λ x <<λ Re x π / λ

24 Surface lasmon 응용 Barnes et al., Nature (003)

25 Metal waveguies Surface lasmonic waveguies Several 전자신호 1 cm long, 15 nm thin an 8 micron wie gol stries guiing LRSPPs 전자신호 3-6 mm long control electroes low riving owers (arox. 100 mw) an high extinction ratios (arox. 30 B) resonse times (arox. 0.5 ms) total (fiber-to-fiber) insertion loss of arox. 금속선광신호 8 B when using single-moe fibers 광신호

26 Asymmetric moe : fiel enhancement at a metallic ti E r E r E z E z M. I. Stocman, Nanofocusing of Otical Energy in Taere Plasmonic Waveguies, Phys. Rev. Lett. 93, (004)]

27 Nano-scale light guiing

28 Metal Nanoarticle Waveguies Maier et al., Av. Mater. 13, 1501 (001)

29 Nano Plasmonics

30 Nano-Photonics Base on Plasmonics By Prof. M. Brongersma Nanoscale integrate circuits with the oerating see of hotonics can be ossible!

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32 A Worl of NanoPlasmonics On-chi light source Long-range(~ cm) waveguies ~ cm Short-range(~ nm) waveguies Photonic integrate circuit Nanohotonics Nano-electronics Harry Atwater, California Institute of Technology Coul such an Architecture be Realize with Metal rather than Dielectric Waveguie Technology?