Ne2E/m Polarization: P=Np=略-w(w+in) =0XeE, Dipole moment p=-er Dielectric susceptibility Ne2/mEo e(w)=1+Xe(w)=1+ w话-w(w+iy) f2 Ne2/meo 6 Oscillator strength Re(E,) Note that if the oscillator --lm(e,） strength is strong and the Re() (3)wl dissipation is small Im() f= 0.03w 2 pue enough,the real part of the permittivity can ⊙ 0 1=0.02500 become negative at 正 -2 frequencies just above the f=0.1娟 resonance frequency =4 0.7 0.8 0.9 1.01.11.21.3 Y2=0.010 00
Polarization: Dipole moment Dielectric susceptibility Oscillator strength 1 0 0.025 2 0 0.01 Note that if the oscillator strength is strong and the dissipation is small enough, the real part of the permittivity can become negative at frequencies just above the resonance frequency
Metal and plasmons at optical frequency The motion for an electron in a time harmonic electric field is: mi+myi =-eEo exp(-iwt), P=(e-1)EoE--Ncr=_ e2/mE (w+iy) plasma frequency e(u)=1- Ne2/(com) w w(w+iy)】 -1- w(w+iy) Evanescence Propagation -Re(k/k。) 。 Im(k/k -Re(E) -Im() k2c2+w2=w2 10 2 3 00p 0/0. Dispersion of light in a good plasma
Metal and plasmons at optical frequency The motion for an electron in a time harmonic electric field is: Dispersion of light in a good plasma
Thin wire structures as low frequency negative permittivity (Pendry 1996) For many metals,the plasma frequency is at ultraviolet frequencies and y is small compared with.However,the dissipation in most metals is very large,so we have trouble when try to extend this behavior to low frequencies. In 1996,Pendry proposed that thin metallic wires behave as a low frequency plasma with a frequency stop-band from zero up to a cutoff frequency.The low frequency stop-band can be attributed to negative dielectric permittivity and the cutoff frequency to the plasma frequency. 个E
For many metals, the plasma frequency is at ultraviolet frequencies and is small compared with . However, the dissipation in most metals is very large, so we have trouble when try to extend this behavior to low frequencies. p Thin wire structures as low frequency negative permittivity (Pendry 1996) In 1996, Pendry proposed that thin metallic wires behave as a low frequency plasma with a frequency stop-band from zero up to a cutoff frequency. The low frequency stop-band can be attributed to negative dielectric permittivity and the cutoff frequency to the plasma frequency. E
Constraining electrons to move along thin wires has two consequences: 1.The average electron density is reduced because only part of space is filled by metal 元r2 neff a2 n, 2.The thin wires have a large self-inductance and it is not easy to change the currents flowing in them.So it appears as if the electrons had acquired a tremendously large mass H=品（日。） The vector potential associated with the field H(p)=67×A aoi=[nn] a yp< a-p =0 p 2 I=πr2nev r<a Approximation in the mean field limit A(p)≈lonr2neu 2π \p】
Constraining electrons to move along thin wires has two consequences: 1. The average electron density is reduced because only part of space is filled by metal 2. The thin wires have a large self-inductance and it is not easy to change the currents flowing in them. So it appears as if the electrons had acquired a tremendously large mass a I 1 0 ( ) H A The vector potential associated with the field Approximation in the mean field limit 0
We note that the canonical momentum of an electron in an electromagnetic field is p +eA.Thus assuming that the electrons flow on the surface of the wire (assuming a perfect conductor),we can associate a momentum per unit length of the wire of mentr2nv and thus an effective mass of uoπr2ne2 meff= 2元 we i() for the electron. Thus,assuming a longitudinal plasmonic mode for the system,we have neme2 2元c2 0p= Eomeff a2In(a/r) Typically one can chooser =1 um,a 10 mmand aluminium wires (n 1029 m-3), which gives an effective mass of mem=2.67×10-26kg, that is almost 15 times that of a proton,and a plasma frequency of about 2 GHz!Thus,we have succeeded in obtaining a negative dielectric material at microwave frequencies