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 𝜌 = 0, and there is no large scale electric field in the plasma.

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Plasma Oscillation

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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”.

Plasma Oscillation contd.

Derivation of Plasma frequency

Derivation of Plasma frequency contd.

Derivation of Plasma frequency

𝑓0= 8.98 𝑁

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,

𝐸 = ℏ𝜔0(= ℎ𝑓0) where 𝜔0, 𝑓0 are plasma angular frequency and plasma frequency.

Dispersion Relation for plasma o Recall the Lorentz Dispersion relation for gaseous medium

𝜖𝑟 𝑜𝑟 𝑛2 = 1 +

𝑁𝑞2

𝜖0𝑚

𝑗

𝑓𝑗

𝜔𝑗2 −𝜔2 − 𝜄 𝛾𝜔

In plasma

i. The electrons are free and not bound to ions by any kind of spring like force, so 𝛽 = 0 ⇒𝜔𝑗 = 0

ii. All electrons experience same force, so no summation required,

iii. Also, q=e , and there is negligible damping in plasma: so, 𝛾 = 0.Thus, we have :

𝜖𝑟 = 𝑛2 = 1 −

𝑁𝑒2

𝜖0𝑚

1

−𝜔2. We know 𝜔0

2 = 𝑁𝑒2/𝑚𝜖0,

Thus,𝜖𝑟 = 𝑛

2 = 1 −𝜔02

𝜔2

• Expression of dielectric constant and refractive index of

plasma.

• This expression is also called the dispersion relation of

plasma.

• It is clear that refractive index of plasma n depends on

the frequency (𝜔) of the wave passing through it.

Surprise in Dispersion relation of plasma

𝜖𝑟 = 𝑛2 = 1 −

𝜔02

𝜔2

In this relation two surprising elements can be seen

1. 𝑛2 < 1𝑚𝑒𝑎𝑛𝑠 𝑐/𝑣 < 1 ⟹ 𝑣 > 𝑐 , Surprising !!!! Is it violation of Einstein’s

special relativity??

2. For 𝜔 < 𝜔0, dielectric constant 𝜖𝑟 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,

we get 𝑣 =𝑐

1−𝜔02

𝜔2

• As, 𝑣 = 𝑣𝑝 = 𝜔/𝑘, substitution of v gives,

𝑘2𝑐2

𝜔2= 1 −

𝜔02

𝜔2⇒

• Above expression is an alternate form of dispersion relation (DR) written in terms of wave vector k and angular freq. 𝜔.• Group velocity is the physical velocity which gives the rate of energy/signal transfer in a medium.

• 𝑣𝑔 =𝑑𝜔

𝑑𝑘: differentiation of DR w.r.t. k gives

2𝑘𝑐2𝑑𝑘 = 2𝜔𝑑𝜔 ⟹𝑑𝜔

𝑑𝑘=𝑘𝑐2

𝜔=𝑐2

𝑣𝑝

⟹ 𝑣𝑔 = 𝑐 1 −𝜔02

𝜔2, now, 𝑣𝑔 < 𝑐 which is cosistant with Relativity

𝑘2𝑐2 = 𝜔2 − 𝜔02

Imaginary Refractive index & Negative Dielectric constant: Physical meaning

𝑘2𝑐2 = 𝜔2 − 𝜔02 ⇒ 𝑘𝑐 = (𝜔2 − 𝜔0

2)

• Wave Equation : 𝐸 = 𝐸0 𝑒𝑖(𝑘𝑧−𝜔𝑡),

• Waves can propagate through a medium only if k has a real part.

• If 𝜔 < 𝜔0, dielectric constant 𝜖𝑟 is negative and refractive index n is pure imaginary. Also 𝑘 = 𝑖 𝑘 i.e. k is also a pure

imaginary number.

• Hence wave equation becomes, 𝐸 = 𝐸0𝑒𝑖(𝑘𝑧−𝜔𝑡) = 𝐸0𝑒

−|𝑘|𝑧𝑒−𝑖𝜔𝑡.

• Here the first term 𝑒−|𝑘|𝑧 is exponentially decaying with space while the second term 𝑒−𝑖𝜔𝑡 gives oscillation with time.

Thus for 𝜔 < 𝜔0, wave take the form of decaying standing waves.

• In fact, for 𝜔 < 𝜔0, the EM wave incident on plasma does not propagate in plasma , instead it will be totally reflected.

(How ??) Answer comes from the EM theory.

• Energy flux of EM wave 𝑢 =𝐸 2

𝜇0𝜔Re(k). For 𝜔 < 𝜔0, i.e. k is pure imaginary, so, 𝑢 = 0 which means NO energy is

transferred in the plasma. Thus wave is completely reflected.

𝜖𝑟 = 𝑛2 = 1 −

𝜔02

𝜔2

o Plasma frequency sets a lower cutoff for the frequencies of electromagnetic radiation.

o Waves of frequencies lower than plasma frequency (𝜔 < 𝜔0), are reflected back by the plasma.

o Only those radiations for which frequency is greater than plasma frequency (𝜔 > 𝜔0), can pass

through a plasma.

o For very high frequencies i.e. 𝜔 ≫ 𝜔0 plasma behaves as a transparent non-dispersive medium.

How??

o In the limit 𝜔 ≫ 𝜔0, 𝜖𝑟⟶ 1, 𝑛 ⟶ 1. DR becomes 𝑘2𝑐2 = 𝜔2, where 𝑣𝑝 = 𝑣𝑔 = 𝑐. Since,

dielectric constant, refractive index and phase velocity is independent of frequency, there is no

dispersion of wave.

Conclusion

𝜖𝑟 = 𝑛2 = 1 −

𝜔02

𝜔2𝑘2𝑐2 = 𝜔2 − 𝜔0

2orDispersion Relation (DR)

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.)

• In Cu, then plasma frequency, fo ≈ 2.6 × 1015 Hz.

• This is higher than the frequencies of visible light, and explains why

metals are opaque to visible light and transparent to UV light.