Optical Properties of Plasma

Course: B.Sc. Physical Sciences Semester: VI, Section C Paper: Solid State Physics Instructor: Manish K. Shukla Plasma • Plasma is a gas of charge particles. • The plasma is overall neutral, i.e., the number density of the electrons and ions are the same. • Under normal conditions, there are always equal numbers positive ions and electrons in any volume of the plasma, so the charge density 휌 = 0, and there is no large scale electric field in the plasma.

22 April 2020 2 Plasma Oscillation

22 April 2020 3 Plasma Oscillation contd.

 Now imagine that all of the electrons are displaced to the right by a small amount x, while the positive ions are held fixed, as shown on the right side of the figure above.  The displacement of the electrons to the right leaves an excess of positive charge on the left side of the plasma slab and an excess of negative charge on the right side, as indicated by the dashed rectangular boxes.  The positive slab on the left and the negative slab on the right produce an electric field pointing toward the right that pulls the electrons back toward their original locations.  However, the electric force on the electrons causes them to accelerate and gain kinetic energy, so they will overshoot their original positions.  This situation is similar to a mass on a horizontal frictionless surface connected to a horizontal spring.  In the present problem,. the electrons execute simple harmonic motion at a frequency that is called the “plasma frequency”. Derivation of Plasma frequency Derivation of Plasma frequency contd. Derivation of Plasma frequency

푓0= 8.98 푁 Plasmon o A plasmon is a quantum of plasma oscillation. o Just as light (an optical oscillation) consists of photons, the plasma oscillation consists of plasmons. o The plasmon can be considered as a quasiparticle since it arises from the quantization of plasma oscillations, just like phonons are quantizations of mechanical vibrations. o Thus, plasmons are collective (a discrete number) oscillations of the free electron gas density. o At optical frequencies, plasmons can couple with a photon to create another quasiparticle called a plasmon polariton. o The plasmon energy can often be estimated in the free electron model as,

퐸 = ℏ휔0(= ℎ푓0) where 휔0, 푓0 are plasma angular frequency and plasma frequency. Dispersion Relation for plasma o Recall the Lorentz Dispersion relation for gaseous medium 2 2 푁푞 푓푗 휖푟 표푟 푛 = 1 + 2 2 휖0푚 휔 − 휔 − 휄 훾휔 푗 푗 In plasma i. The electrons are free and not bound to ions by any kind of spring like force, so 훽 = 0 ⇒ 휔푗 = 0 ii. All electrons experience same force, so no summation required, iii. Also, q=e , and there is negligible damping in plasma: so, 훾 = 0. Thus, we have : 2 2 푁푒 1 2 2 휖푟 = 푛 = 1 − 2 . We know 휔0 = 푁푒 /푚휖0, 휖0푚 −휔 • Expression of dielectric constant and refractive index of 2 plasma. Thus, 2 휔0 휖 = 푛 = 1 − • 푟 휔2 This expression is also called the dispersion relation of plasma. • It is clear that refractive index of plasma n depends on the frequency (휔) of the wave passing through it. Surprise in Dispersion relation of plasma In this relation two surprising elements can be seen 휔2 휖 = 푛2 = 1 − 0 1. 푛2 < 1 푚푒푎푛푠 푐/푣 < 1 ⟹ 푣 > 푐, Surprising !!!! Is it violation of Einstein’s 푟 휔2 special relativity??

2. For 휔 < 휔0, dielectric constant 휖푟 is negative and refractive index n is pure imaginary. • In the definition of n=c/v, v is in fact phase velocity of EM wave in a given medium. In above expression, putting n=c/v, 푐 we get 푣 = 휔2 1− 0 휔2 • As, 푣 = 푣푝 = 휔/푘, substitution of v gives, 2 2 2 푘 푐 휔 2 = 1 − 0 ⇒ 푘2푐2 = 휔2 − 휔 휔2 휔2 0 • Above expression is an alternate form of dispersion relation (DR) written in terms of wave vector k and angular freq. 휔. • Group velocity is the physical velocity which gives the rate of energy/signal transfer in a medium. 푑휔 • 푣 = : differentiation of DR w.r.t. k gives 푔 푑푘 푑휔 푘푐2 푐2 2푘푐2푑푘 = 2휔푑휔 ⟹ = = 푑푘 휔 푣푝 휔2 ⟹ 푣 = 푐 1 − 0 , now, 푣 < 푐 which is cosistant with Relativity 푔 휔2 푔 Imaginary Refractive index & Negative Dielectric constant: Physical meaning

2 2 휔0 2 2 2 2 2 2 휖 = 푛 = 1 − 푘 푐 = 휔 − 휔0 ⇒ 푘푐 = (휔 − 휔0) 푟 휔2

푖(푘푧−휔푡) • Wave Equation : 퐸 = 퐸0 푒 , • Waves can propagate through a medium only if k has a real part.

• If 휔 < 휔0, dielectric constant 휖푟 is negative and refractive index n is pure imaginary. Also 푘 = 𝑖 푘 i.e. k is also a pure imaginary number. 푖(푘푧−휔푡) −|푘|푧 −푖휔푡 • Hence wave equation becomes, 퐸 = 퐸0푒 = 퐸0푒 푒 . • Here the first term 푒−|푘|푧 is exponentially decaying with space while the second term 푒−푖휔푡 gives oscillation with time.

Thus for 휔 < 휔0, wave take the form of decaying standing waves.

• In fact, for 휔 < 휔0, the EM wave incident on plasma does not propagate in plasma , instead it will be totally reflected. (How ??) Answer comes from the EM theory.

퐸 2 • Energy flux of EM wave 푢 = Re(k). For 휔 < 휔0, i.e. k is pure imaginary, so, 푢 = 0 which means NO energy is 휇0휔 transferred in the plasma. Thus wave is completely reflected. Conclusion 2 휔0 2 2 2 2 Dispersion Relation 휖 = 푛2 = 1 − or 푘 푐 = 휔 − 휔0 푟 휔2 (DR) o Plasma frequency sets a lower cutoff for the frequencies of electromagnetic radiation. o Waves of frequencies lower than plasma frequency (휔 < 휔0), are reflected back by the plasma. o Only those radiations for which frequency is greater than plasma frequency (휔 > 휔0), can pass through a plasma. o For very high frequencies i.e. 휔 ≫ 휔0 plasma behaves as a transparent non-dispersive medium. How??

2 2 2 o In the limit 휔 ≫ 휔0, 휖푟 ⟶ 1, 푛 ⟶ 1. DR becomes 푘 푐 = 휔 , where 푣푝 = 푣푔 = 푐. Since, dielectric constant, refractive index and phase velocity is independent of frequency, there is no dispersion of wave. Application of Plasma : Ionosphere Application of plasma II : Metals • The metals shine by reflecting most of light in visible range. • The concept of electron plasma oscillations can also be applied to the free electrons in a conductor. • The visible light can't pass through the metal because the plasma frequency of electrons in metal falls in ultraviolet region. • For frequencies in UV, metals are transparent. • For example, the free electron density in Cu is 8.4 × 1028 m−3. (By way of comparison, the density of air molecules at a pressure of one atmosphere and T = 300K is 2.4 × 1025 m−3.) 15 • In Cu, then plasma frequency, fo ≈ 2.6 × 10 Hz. • This is higher than the frequencies of visible light, and explains why metals are opaque to visible light and transparent to UV light.