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Waves in Isotropic Cold Plasma: Dispersive Medium*

9
Waves in Isotropic Cold Plasma: Dispersive Medium *

In Section 8.4, we briefly discussed the plasma medium and characterized it by the associated parameter called the plasma frequency ω_p, which is related to the electron density N through Equation 8.22. In Chapter 10, we will discuss quantitatively the various technical terms used in describing the plasma state of matter.

In this chapter, we consider the wave propagation in an isotropic cold (thermal effects neglected) plasma medium. We start with the equation for the force experienced by the electron in the plasma medium due to the wave electric and magnetic fields, given by

(9.1a)

$m \frac{d v}{d t} = - q [E + v \times B] .$

It can be shown that the force due to the wave magnetic field is negligible compared to the force due to the wave electric field.

The plasma current density J is related to its density N by

(9.1b)

$J = - q N v$

After neglecting the ν × B term in (9.1a), and using (8.22) and (9.1b), one can obtain the equation relating J and E:

(9.1c)

$\frac{d J}{d t} = ε_{0} ω_{p}^{2} E$

Let us assume that the electron density varies only in space (inhomogeneous isotropic plasma):

(9.1d)

$ω_{p}^{2} = ω_{p}^{2} (r) .$

9.1Basic Equations

One can then construct basic solutions by assuming that the field variables vary harmonically in time:

(9.2)

$F (r, t) = F (r) exp (j ω t) .$

In the above, F(r) stands for any of E, H, or J phasors. The basic field equations are then given by

(9.3)

$\nabla \times E = - j ω μ_{0} H,$

(9.4)

$\nabla \times H = j ω ε_{0} E + J,$

(9.5)

$j ω J = ε_{0} ω_{p}^{2} (r) E .$

Combining Equations 9.4 and 9.5, we can write

(9.6)

$\nabla \times H = j ω ε_{0} ε_{p} (r, ω) E,$

where ε_p is the dielectric constant of the isotropic plasma and is given by

(9.7)

$ε_{p} (r, ω) = 1 - \frac{ω_{p}^{2} (r)}{ω^{2}} .$

The vector wave equations are

(9.8)

$\nabla^{2} E + \frac{ω^{2}}{c^{2}} ε_{p} (r) E = \nabla (\nabla \cdot E),$

(9.9)

$\begin{array}{c} \nabla^{2} H + \frac{ω^{2}}{c^{2}} ε_{p} (r) H & = & - \frac{ε_{0}}{j ω} \nabla ω_{p}^{2} (r) \times E \\ = & - j ω ε_{0} \nabla ε_{p} (r) \times E . \end{array}$

The scalar one-dimensional equations take the form

(9.10)

$\frac{\partial E}{\partial z} = - j ω μ_{0} H,$

(9.11)

$- \frac{\partial H}{\partial z} = j ω ε_{0} E + J,$

(9.12)

$j ω J = ε_{0} ω_{p}^{2} (z) E .$

Combining Equations 9.11 and 9.12, or from Equations 9.6, we have

(9.13)

$- \frac{\partial H}{\partial z} = j ω ε_{0} ε_{p} (z, ω) E,$

where

(9.14)

$ε_{p} (z, ω) = 1 - \frac{ω_{p}^{2} (z)}{ω^{2}} .$

Figure 9.1 shows the variation of ε_p with ω. ε_p is negative for ω < ω_p, 0 for ω = ω_p, and positive but less than 1 for ω > ω_p. The scalar one-dimensional wave equations, in this case, reduce to Equations 9.15 and 9.16:

(9.15)

$\frac{d^{2} E}{d z^{2}} + \frac{ω^{2}}{c^{2}} ε_{p} (z, ω) E = 0,$

(9.16)

$\frac{d^{2} H}{d z^{2}} + \frac{ω^{2}}{c^{2}} ε_{p} (z, ω) H = \frac{1}{ε_{p} (z, ω)} \frac{d ε_{p} (z, ω)}{d z} \frac{d H}{d z} .$

FIGURE 9.1
Dielectric constant versus frequency for a plasma medium.

Equation 9.15 can easily be solved if we further assume that the dielectric is homogeneous (ε_p is not a function of z). Such a solution is given by

(9.17)

$E = E^{+} (z) + E^{-} (z),$

(9.18)

$E = E_{0}^{+} exp (- j k z) + E_{0}^{-} exp (+ j k z),$

where

(9.19)

$k = \frac{ω}{c} \sqrt{ε_{p}} = k_{0} \sqrt{ε_{p}} = k_{0} n .$

Here, n is the refractive index and k₀ is the free-space wave number. The first term on the RHS of Equation 9.17 represents a traveling wave in the positive z-direction (positive-going wave) of angular frequency ω and wave number k and the second term a similar but negative-going wave. The z-direction is the direction of phase propagation, and the velocity of phase propagation v_p (phase velocity) of either of the waves is given by

(9.20)

$v_{p} = \frac{ω}{k} .$

From Equation 9.10,

(9.21)

$H^{+} (z) = \frac{1}{(- j ω μ_{0})} (- j k) E_{0}^{+} exp (- j k z) = \frac{1}{η} E^{+} (z),$

(9.22)

$H^{-} (z) = - \frac{1}{η} E^{-} (z),$

where

(9.23)

$η = \sqrt{\frac{μ_{0}}{ε_{0} ε_{p}}} = \frac{η_{0}}{n} = \frac{120 π}{n} (Ω) .$

Here η is the intrinsic impedance of the medium. The waves described above which are one-dimensional solutions of the wave equation in an unbounded isotropic homogeneous medium are called uniform plane waves in the sense that the phase and the amplitude of the waves are constant in a plane. Such one-dimensional solutions in a coordinate-free description are called transverse electric and magnetic waves (TEM) and for an arbitrarily directed wave their properties can be summarized as follows:

(9.24)

$\hat{E} \times \hat{H} = \hat{k},$

(9.25)

$\hat{E} \cdot \hat{k} = 0, \hat{H} \cdot \hat{k} = 0,$

(9.26)

$E = η H .$

In the above, $\hat{k}$ is a unit vector in the direction of phase propagation. In other words, the properties are:

Unit electric field vector, unit magnetic field vector, and the unit vector in the direction of (phase) propagation form a mutually orthogonal system.
There is no component of the electric or the magnetic field vector in the direction of propagation.
The ratio of the electric field amplitude to the magnetic field amplitude is given by the intrinsic impedance of the medium.

Figure 9.2 shows the variation of the dielectric constant ε_p with frequency for a typical real dielectric. For such a medium, in a broad frequency band, ε_p can be treated as not varying with ω and denoted by the dielectric constant ε_r. We will next consider the step profile approximation for a dielectric profile. In each of the media 1 and 2, the dielectric can be treated as homogeneous. The solution for this problem is well known but is included here to provide a comparison with the solutions in Chapter 10, Chapter 11 and Chapter 12.

FIGURE 9.2
Sketch of the dielectric constant of a typical material.

9.2Dielectric–Dielectric Spatial Boundary

The geometry of the problem is shown in Figure 9.3. Let the incident wave in medium 1, also called the source wave, have the fields

(9.27)

$E_{i} (x, y, z, t) = \hat{x} E_{0} exp [j (ω_{i} t - k_{i} z)],$

(9.28)

$H_{i} (x, y, z, t) = \hat{y} \frac{E_{0}}{η_{1}} exp [j (ω_{i} t - k_{i} z)],$

where ω_i is the frequency of the incident wave and

(9.29)

$k_{i} = ω_{i} \sqrt{μ_{0} ε_{1}} = \frac{ω_{i}}{v_{p 1}} .$

FIGURE 9.3
Dielectric–dielectric spatial boundary problem.

The fields of the reflected wave are given by

(9.30)

$E_{r} (x, y, z, t) = \hat{x} E_{r} exp [j (ω_{r} t + k_{r} z)],$

(9.31)

$H_{r} (x, y, z, t) = - \hat{y} \frac{E_{r}}{η_{1}} exp [j (ω_{r} t + k_{r} z)],$

where ω_r is the frequency of the reflected wave and

(9.32)

$k_{r} = ω_{r} \sqrt{μ_{0} ε_{1}} .$

The fields of the transmitted wave are given by

(9.33)

$E_{t} (x, y, z, t) = \hat{x} E_{t} exp [j (ω_{t} t - k_{t} z)],$

(9.34)

$H_{t} (x, y, z, t) = \hat{y} \frac{E_{t}}{η_{2}} exp [j (ω_{t} t - k_{t} z)],$

where ω_r is the frequency of the transmitted wave and

(9.35)

$k_{t} = ω_{t} \sqrt{μ_{0} ε_{2}} = \frac{ω_{t}}{v_{p 2}} .$

The boundary condition of the continuity of the tangential component of the electric field at the interface z = 0 can be stated as

(9.36)

$E (x, y, 0^{-}, t) \times \hat{z} = E (x, y, 0^{+}, t) \times \hat{z},$

(9.37)

$[E_{i} (x, y, 0^{-}, t) + E_{r} (x, y, 0^{-}, t)] \times \hat{z} = [E_{t} (x, y, 0^{+}, t)] \times \hat{z} .$

The above must be true for all x, y, and t. Thus, we have

(9.38)

$E_{0} exp [j ω_{i} t] + E_{r} exp [j ω_{r} t] = E_{t} exp [j ω_{t} t] .$

Since Equation 9.38 must be satisfied for all t, the coefficients of t in the exponents of Equation 9.38 must match:

(9.39)

$ω_{i} = ω_{r} = ω_{t} = ω .$

The above result can be stated as follows: the frequency ω is conserved across a spatial discontinuity in the properties of an electromagnetic medium. As the wave crosses from one medium to the other in space, the wave number k changes as dictated by the change in the phase velocity (see Figure 9.4).

FIGURE 9.4
Conservation of frequency across a spatial boundary.

From Equations 9.38 and 9.39, we obtain

(9.40)

$E_{0} + E_{r} = E_{t} .$

The second independent boundary condition of continuity of the tangential magnetic field component is written as

(9.41)

$[H_{i} (x, y, 0^{-}, t) + H_{r} (x, y, 0^{-}, t)] \times \hat{z} = [H_{t} (x, y, 0^{+}, t)] \times \hat{z} .$

The reflection coefficient R_A = E_r/E₀ and the transmission coefficient T_A = E_t/E₀ are determined using Equations 9.40 and 9.41. From Equation 9.41, we have

(9.42)

$\frac{E_{0} - E_{r}}{η_{1}} = \frac{E_{t}}{η_{2}} .$

The results are as follows:

(9.43)

$R_{A} = \frac{η_{2} - η_{1}}{η_{2} + η_{1}} = \frac{n_{1} - n_{2}}{n_{1} + n_{2}},$

(9.44)

$T_{A} = \frac{2 η_{2}}{η_{2} + η_{1}} = \frac{2 n_{1}}{n_{1} + n_{2}} .$

The significance of the subscript A in the above is to distinguish from the reflection and transmission coefficients due to a temporal discontinuity in the properties of the medium.

We next show that the time-averaged power density of the source wave is equal to the sum of the time-averaged power density of the reflected and the transmitted waves:

(9.45)

$|\frac{1}{2} Re [E_{i} \times H_{i}^{*} \cdot \hat{z}]| = |\frac{1}{2} Re [E_{r} \times H_{r}^{*} \cdot \hat{z}]| + |\frac{1}{2} Re [E_{t} \times H_{t}^{*} \cdot \hat{z}]| .$

The LHS is

(9.46)

$\frac{1}{2} E_{0} H_{0} = \frac{1}{2} \frac{E_{0}^{2}}{η_{1}},$

whereas the RHS is

(9.47)

$\frac{1}{2} E_{0}^{2} [\frac{{|R_{A}|}^{2}}{η_{1}} + \frac{{|T_{A}|}^{2}}{η_{2}}] = \frac{1}{2} \frac{E_{0}^{2}}{η_{1}} [\frac{{(η_{2} - η_{1})}^{2} + 4 η_{1} η_{2}}{{(η_{2} + η_{1})}^{2}}] = \frac{1}{2} \frac{E_{0}^{2}}{η_{1}} .$

9.3Reflection by a Plasma Half-Space

Let an incident wave of frequency ω traveling in the free space (z < 0) be incident normally on the plasma half-space (z > 0) of plasma frequency ω_p. The intrinsic impedance of the plasma medium is given by

(9.48)

$η_{p} = \frac{η_{0}}{n_{p}} = \frac{120 π ω}{\sqrt{ω^{2} - ω_{p}^{2}}} .$

From Equation 9.43, the reflection coefficient R_A is given by

(9.49)

$R_{A} = \frac{η_{p} - η_{0}}{η_{p} + η_{0}} = \frac{1 - n_{p}}{1 + n_{p}} = \frac{Ω - \sqrt{Ω^{2} - 1}}{Ω + \sqrt{Ω^{2} - 1}},$

where

(9.50)

$Ω = \frac{ω}{ω_{p}}$

is the source frequency normalized with respect to the plasma frequency. The power reflection coefficient ρ = |R_A|² and the power transmission coefficient τ (=1 − ρ) versus Ω are shown in Figure 9.5. In the frequency band 0 < ω < ω_p, the characteristic impedance η_p is imaginary. The electric field E_P and the magnetic field H_P in the plasma are in time quadrature; the real part of $(E_{P} \times H_{P}^{*})$ is zero. The wave in plasma is an evanescent wave [1] and carries no real power. The source wave is totally reflected and ρ = 1. In the frequency band ω_p < ω < ∞, the plasma behaves as a dielectric and the source wave is partially transmitted and partially reflected. The time-averaged power density of the incident wave is equal to the sum of the time-averaged power densities of the reflected and transmitted waves.

FIGURE 9.5
Sketch of the power reflection coefficient (ρ) and the power transmission coefficient (τ) of a plasma half-space.

Metals at optical frequencies are modeled as plasmas with plasma frequency ω_p of the order of 10¹⁶. Refining the plasma model by including the collision frequency will lead to three frequency domains of conducting, cutoff, and dielectric phenomena [2]. A more comprehensive account of modeling metals as plasmas is given in References [3,4]. The associated phenomena of attenuated total reflection, surface plasmons, and other interesting topics are based on modeling metals as plasmas. A brief account of surface wave is given in Section 9.8.

9.4Reflection by a Plasma Slab [5]

In this section, we will consider oblique incidence to add to the variety to the problem formulation. The geometry of the problem is shown in Figure 9.6. Let x–z be the plane of incidence and $\hat{k}$ be the unit vector along the direction of propagation. Let the magnetic field H^I be along y and the electric field E^I lie entirely in the plane of incidence. Such a wave is described in the literature by various names: (1) p wave, (2) TM wave, and (3) parallel-polarized wave. Since the boundaries are at z = 0 and d, it is necessary to use x–y–z-coordinates in formulating the problem and express $\hat{k}$ in terms of x- and z-coordinates. The problem, though one-dimensional, appears to be two-dimensional. Equation 9.51, Equation 9.52, Equation 9.53, Equation 9.54, Equation 9.55 and Equation 9.56 describe the incident wave:

(9.51)

$H^{I} = \hat{y} H_{y}^{I} exp [- j k_{0} \hat{k} \cdot r],$

(9.52)

$\hat{k} = \hat{x} S + \hat{z} C,$

(9.53)

$S = sin θ_{I}, C = cos θ_{I},$

(9.54)

$E^{I} = (\hat{y} \times \hat{k}) E^{I} exp [- j k_{0} \hat{k} \cdot r] = (\hat{x} E_{x}^{I} + \hat{z} E_{z}^{I}) Ψ^{I},$

(9.55)

$E_{x}^{I} = C E^{I}, E_{z}^{I} = - S E^{I}, E^{I} = η_{0} H_{y}^{I},$

(9.56)

$Ψ^{I} = exp [- j k_{0} (S x + C z)] .$

The reflected wave is written as

(9.57)

$E^{R} = (\hat{x} E_{x}^{R} + \hat{z} E_{z}^{R}) Ψ^{R},$

(9.58)

$H^{R} = \hat{y} H_{y}^{R} Ψ^{R},$

(9.59)

$Ψ^{R} = exp [- j k_{0} (S x - C z)] .$

The boundary conditions at z = 0 require that the x dependence in the region z > 0 remains the same as in the region z < 0. Since the region z > d is also a free space, the exponential factor for the transmitted wave will be the same as the incident wave and the fields of the transmitted wave are

(9.60)

$E^{T} = (\hat{x} E_{x}^{T} + \hat{z} E_{z}^{T}) Ψ^{T},$

(9.61)

$H^{T} = \hat{y} H_{y}^{T} Ψ^{T},$

(9.62)

$Ψ^{T} = exp [- j k_{0} (S x + C z)] .$

All the field amplitudes in the free space can be expressed in terms of $E_{x}^{I}$ , $E_{x}^{R}$ , or $E_{x}^{T}$ :

(9.63)

$C E_{z}^{I, T} = - S E_{x}^{I, T},$

(9.64)

$C E_{z}^{R} = S E_{x}^{R},$

(9.65)

$C η_{0} H_{y}^{I, T} = E_{x}^{I, T},$

(9.66)

$C η_{0} H_{y}^{R} = - E_{x}^{R} .$

We consider next the waves in the homogeneous plasma region 0 < z < d. The x and the z dependence for waves in the plasma comes from the exponential factor Ψ^P:

(9.67)

$Ψ^{P} = exp [- j k_{0} (S x + q z)],$

where q has to be determined. The wave number in the plasma is given as $k_{p} = k_{0} \sqrt{ε_{p}} .$ Thus, we have

(9.68)

$q^{2} + S^{2} = ε_{p} = 1 - \frac{ω_{p}^{2}}{ω^{2}},$

(9.69)

$q_{1, 2} = \pm \sqrt{C^{2} - \frac{ω_{p}^{2}}{ω^{2}}} = \pm \sqrt{\frac{C^{2} Ω^{2} - 1}{Ω^{2}}} .$

The two values for q indicate the excitation of two waves in the plasma, the first is the positive wave and the second is a negative wave. For Ω < 1/C, q is imaginary and the two waves in the plasma are evanescent. The next section deals with this frequency band.

The fields in the plasma can be written as

(9.70)

$E^{P} = \sum_{m = 1}^{2} (\hat{x} E_{x m}^{P} + \hat{z} E_{z m}^{P}) Ψ_{m}^{P},$

(9.71)

$H^{P} = \sum_{m = 1}^{2} \hat{y} H_{y m}^{P} Ψ_{m}^{P},$

(9.72)

$Ψ_{m}^{P} = exp [- j k_{0} (S x + q_{m} z)] .$

All the field amplitudes in the plasma can be expressed in terms of $E_{x 1}^{P}$ and $E_{x 2}^{P}$ :

(9.73)

$η_{0} H_{y m}^{P} = \frac{ε_{p}}{q_{m}} E_{x m}^{P} = η_{y m} E_{x m}^{P},$

(9.74)

$E_{z m}^{P} = - \frac{S}{q_{m}} E_{x m}^{P} .$

The above relations are obtained from Equations 9.3 and 9.6 by noting ∂/∂x = − jk₀S and ∂/∂z = − jk₀q; they can also be written from inspection. Assuming that $E_{x}^{I}$ is known, the unknowns reduce to four, that is, $E_{x 1}^{P}$ , $E_{x 2}^{P}$ , $E_{x}^{R}$ , and $E_{x}^{T}$ . They can be determined from the four boundary conditions of continuity of the tangential components E_x and H_y at z = 0 and z = d. In matrix form,

(9.75)

$[\begin{matrix} 1 & 1 & - 1 & 0 \\ C η_{y 1} & C η_{y 2} & 1 & 0 \\ λ_{1} & λ_{2} & 0 & - 1 \\ λ_{1} C η_{y 1} & λ_{2} C η_{y 2} & 0 & - 1 \end{matrix}] [\begin{matrix} E_{x 1}^{P} \\ E_{x 2}^{P} \\ E_{x}^{R} \\ E_{x}^{T} \end{matrix}] = [\begin{matrix} 1 \\ 1 \\ 0 \\ 0 \end{matrix}] E_{x}^{I},$

where

(9.76)

$\begin{array}{c} λ_{1} = exp [j k_{0} (C - q_{1}) d], \\ λ_{2} = exp [j k_{0} (C - q_{2}) d] . \end{array}$

Solving Equation 9.75,

(9.77)

$E_{x 1}^{P} = 2 λ_{2} (1 - C η_{y 2}) E_{x}^{I} / Δ,$

(9.78)

$E_{x 2}^{P} = - 2 λ_{1} (1 - C η_{y 1}) E_{x}^{I} / Δ,$

(9.79)

$E_{x}^{R} = (1 - C η_{y 1}) (1 - C η_{y 2}) (λ_{2} - λ_{1}) E_{x}^{I} / Δ,$

(9.80)

$E_{x}^{T} = 2 λ_{1} λ_{2} C (η_{y 1} - η_{y 2}) E_{x}^{I} / Δ,$

where

(9.81)

$Δ = λ_{2} (1 + C η_{y 1}) (1 - C η_{y 2}) - λ_{1} (1 - C η_{y 1}) (1 + C η_{y 2}) .$

The power reflection coefficient $ρ = {|E_{x}^{R} / E_{x}^{I}|}^{2}$ is given by

(9.82)

$ρ = \frac{1}{1 + {(2 C ε_{p} q_{1} / (C^{2} ε_{p}^{2} - q_{1}^{2}) cosec (2 π Ω q_{1} d_{p}))}^{2}},$

where

(9.83)

$d_{p} = \frac{d}{λ_{p}},$

(9.84)

$λ_{p} = \frac{2 π c}{ω_{p}} .$

In the above, λ_p is the free space wavelength corresponding to the plasma frequency and is used to normalize the slab width. Substituting for q₁, in the range of real q₁,

(9.85)

$ρ = \frac{1}{1 + B {cosec}^{2} A}, \frac{1}{C} < Ω < \infty,$

and in the range of imaginary q₁,

(9.86)

$ρ = \frac{1}{1 - B {cosech}^{2} |A|}, 0 < Ω < \frac{1}{C},$

where A and B are given by

(9.87)

$A = 2 π Ω q_{1} d_{p} = 2 π \sqrt{C^{2} Ω^{2} - 1} d_{p},$

(9.88)

$B = \frac{4 C^{2} ε_{p}^{2} q_{1}^{2}}{{(C^{2} ε_{p}^{2} - q_{1}^{2})}^{2}} = \frac{4 C^{2} Ω^{2} {(Ω^{2} - 1)}^{2} (C^{2} Ω^{2} - 1)}{{(2 C^{2} Ω^{2} - Ω^{2} - C^{2})}^{2}} .$

The power transmission coefficient τ = (1 − ρ) and is given by

(9.89)

$τ = \frac{|B|}{|B| + {sin}^{2} |A|}, \frac{1}{C} < Ω < \infty,$

(9.90)

$τ = \frac{|B|}{|B| + {sinh}^{2} |A|}, 0 < Ω < \frac{1}{C} .$

From Equation 9.85, ρ = 0 when sin A = 0 or

(9.91)

$A = n π, n = 0, 1, 2, \dots .$

There is one more value of Ω for which ρ = 0. From Equation 9.88, when Cε_p = q₁, B = ∞, and from Equation 9.89, τ = 1 and ρ = 0. This point corresponds to a frequency Ω_B given by

(9.92)

$Ω_{B}^{2} = \frac{C^{2}}{2 C^{2} - 1} .$

It is easily shown that Ω_B exists only for the p wave and is greater than 1/C. In fact, this point corresponds to the Brewster angle [4]. Figure 9.7 shows Ω_B versus cos θ_B, where θ_B is the Brewster angle.

FIGURE 9.7
Brewster angle for a plasma medium.

Thus, in the frequency band Ω > 1/C, the variation of ρ with the slab width is oscillatory. Stratton [6] discussed these oscillations for a dielectric slab and associated them with the interference of the internally reflected waves in the slab. In the case of the plasma slab, the variation of ρ with the source frequency is also oscillatory, since in this range the plasma behaves like a dispersive dielectric. The maxima of ρ are less than or equal to 1 and occur for Ω satisfying the transcendental equation

(9.93)

$tan A = f A,$

where

(9.94)

$f = \frac{C^{2} Ω^{2} (Ω^{2} - 1) (C^{2} + Ω^{2} - 2 C^{2} Ω^{2})}{(2 C^{2} Ω^{4} - 2 C^{2} Ω^{2} + C^{2} - Ω^{2}) (1 + Ω^{2} - 2 C^{2} Ω^{2})} .$

Figure 9.8 shows ρ versus Ω for a normalized slab width d_p = 1.0 and C = 0.5 (θ_I = 60°). The inset shows the variation of q with Ω. The oscillations can be clearly seen in the real range of q. Heald and Wharton [2] show similar curves for normal incidence on a lossy plasma slab using parameters normalized with reference to the source wave quantities.

FIGURE 9.8
ρ versus Ω for isotropic plasma slab (parallel polarization).

In Figure 9.8, ρ ≈ 1 in the imaginary range of q showing total reflection of the source wave. However, it will be shown that there can be a considerable tunneling of power for sufficiently thin plasma slabs.

9.5Tunneling of Power through a Plasma Slab [5, 7]

For the frequency band Ω < 1/C, the characteristic roots are imaginary and the waves excited in the plasma are evanescent. The incident wave is completely reflected by the semi-infinite plasma. However, in the case of a plasma slab, some power gets transmitted through it even in this frequency band. It can be shown that this tunneling effect is due to the interaction of the electric field of the positive wave with the magnetic field of the negative wave and vice versa.

The above statement is supported by examining the power flow through the Poynting vector calculation. From Equation 9.77, Equation 9.78, Equation 9.79, Equation 9.80 and Equation 9.81, one can obtain

(9.95)

$E_{x}^{T} = 2 C ε_{p} q_{1} exp [j k_{0} C d] E_{x}^{I} / Δ,$

(9.96)

$E_{x 1}^{P} = (q_{1} + C ε_{p}) q_{1} exp [j k_{0} q_{1} d] E_{x}^{I} / Δ,$

(9.97)

$E_{x 2}^{P} = - (- q_{1} + C ε_{p}) q_{1} exp [- j k_{0} q_{1} d] E_{x}^{I} / Δ,$

where

(9.98)

$Δ = 2 q_{1} C ε_{p} cos (k_{0} q_{1} d) + j (q_{1}^{2} + C^{2} ε_{p}^{2}) sin (k_{0} q_{1} d) .$

The magnetic field components $H_{y 1}^{P}$ , $H_{y 2}^{P}$ , and $H_{y}^{T}$ can be obtained from Equations 9.65 and 9.73.

The power crossing any plane in the plasma parallel to the interface can be found by calculating the z-component of the Poynting vector associated with the plasma waves. This is given by

(9.99)

$S_{z}^{P} = \frac{1}{2} Re [(E_{x 1}^{P} Ψ_{1}^{P} + E_{x 2}^{P} Ψ_{2}^{P}) (H_{y 1}^{P *} Ψ_{1}^{P*} + H_{y 2}^{P*} Ψ_{2}^{P*})],$

which on expansion gives

(9.100)

$\begin{array}{l} S_{z}^{P} & = & \frac{1}{2} Re [E_{x 1}^{P} Ψ_{1}^{P} H_{y 1}^{P^{*}} Ψ_{1}^{P^{*}}] + \frac{1}{2} Re [E_{x 2}^{P} Ψ_{2}^{P} H_{y 1}^{P^{*}} Ψ_{1}^{P^{*}}] \\ + \frac{1}{2} Re [E_{x 1}^{P} Ψ_{1}^{P} H_{y 2}^{P^{*}} Ψ_{2}^{P^{*}}] + \frac{1}{2} Re [E_{x 2}^{P} Ψ_{2}^{P} H_{y 2}^{P^{*}} Ψ_{2}^{P^{*}}] . \end{array}$

where $Ψ_{1}^{P}$ and $Ψ_{2}^{P}$ are given by Equation 9.72. The contribution from each of the four terms on the RHS of Equation 9.100 is now discussed for the two cases of q₁ real Ω > 1/C and q₁ imaginary Ω < 1/C. It is to be noted that the second and the third terms give the contributions due to the cross-interaction of the fields in the positive and negative waves.

Case 1: q₁ real

It is easy to see that the second and the third terms being equal but opposite in sign, there is no contribution to the net power flow from the cross-interaction. The sum of the first and fourth terms is equal to $(1 / 2) Re [(E_{x}^{T} Ψ^{T}) (H_{y}^{T*} Ψ^{T*})]$ , where Ψ^T is given by Equation 9.62. Thus, the power crossing any plane parallel to the slab interface gives exactly the power that emerges into the free space at the other boundary of the slab.

Case 2: q₁ imaginary

It can be shown that each of the first and the fourth terms is zero. This is because the corresponding electric and magnetic fields are in time quadrature. Furthermore, the second term is equal to the third term and each is equal to $(1 / 4) Re [(E_{x}^{T} Ψ^{T}) (H_{y}^{T*} Ψ^{T*})]$ . Thus, the power flow in the tunneling frequency band (Ω < 1/C) comes entirely from the cross-interaction.

At Ω = 1, ε_p = 0, and from Equation 9.88, B = 0 but A ≠ 0. From Equation 9.89, τ = 0. This point exists only for the p wave.

At Ω = 1/C, A = 0, and B = 0. Evaluating the limit, we get the power transmission coefficient τ_c at this point:

(9.101)

$τ_{c} = \frac{1}{1 + S^{4} π^{2} d_{p}^{2}}, Ω = \frac{1}{C} .$

Numerical results of τ versus Ω are presented in Figures 9.9 and 9.10 for the tunneling frequency band. In Figure 9.9, the angle of incidence is taken to be 60° and the curves are given for various d_p. Between Ω = 0 and 1, there is a value of Ω = Ω_max where the transmission is maximum. The maximum value of the transmitted power decreases as d_p increases.

FIGURE 9.9
Transmitted power for an isotropic plasma slab (parallel polarization) in the tunneling range (Ω < 1/C) for various d_p. The right side of the bottom two curves can be seen more clearly than the left side and they are for d_p = 0.5 and d_p = 1.0.

FIGURE 9.10
Transmitted power for an isotropic plasma slab (parallel polarization) in the tunneling range (Ω < 1/C) for various angles of incidence. The broken line curve is for θ_I = 15°.

In Figure 9.10, τ versus Ω is shown with the angle of incidence as the parameter. It is seen that the point of maximum transmission moves to the right as the angle of incidence decreases and merges with Ω = 1 for θ_I = 0° (normal incidence). The maximum value of the transmitted power decreases as θ_I increases. This point is given by the solution of the transcendental equation

(9.102)

$tanh |A| = f |A|,$

where f and A are given in Equations 9.94 and 9.87, respectively. It is suggested that by measuring Ω_max experimentally, one can determine either the plasma frequency or the slab width, knowing the angle of incidence.

9.6Inhomogeneous Slab Problem

Let us next consider the case where ε_p is a function of z in the region 0 < z < d. To simplify and focus on the effect of the inhomogeneity of the properties of the medium, let us revert to the normal incidence. The differential equation for the electric field is given by Equation 9.15. This equation cannot be solved exactly for a general ε_p(z) profile.

However, there are a small number of profiles for which exact solutions can be obtained in terms of special functions. As an example, the problem of the linear profile can be solved in terms of Airy functions. An account of such solutions for a dielectric profile ε_r(z) are given in [8,9] and for a plasma profile ε_p(z, ω) in [2,10]. Abramowitz and Stegun [11] give detailed information on special functions.

A vast amount of literature is available on the approximate solution for an inhomogeneous media problem. The following techniques are emphasized in this book. The actual profile is considered as a perturbation of a profile for which the exact solution is known. The effect of the perturbation is calculated through the use of Green’s function. The problem of a fast profile with finite range—length can be so handled by using a step profile as the reference. At the other end of the approximation scale, the slow profile problem can be handled by adiabatic and WKBJ techniques. See [1,8] for dielectric profile examples and [2,10] for plasma profile examples. Numerical approximation techniques can also be used which are extremely useful in obtaining specific numbers for a given problem and in validating the theoretical results obtained from the physical approximations.

9.7Periodic Layers of Plasma

Next, we consider a particular case of an inhomogeneous plasma medium which is periodic. Let the dielectric function

(9.103)

$ε_{p} (z, ω) = ε_{p} (z + m L, ω),$

where m is an integer and L is the spatial period. If the domain of m is all integers, −∞ < m < ∞, then we have an unbounded periodic media. The solution of Equation 9.15 for the infinite periodic structure problem [1,9,12] will be such that the electric field E(z) differs from the electric field at E(z + L) by a constant:

(9.104)

$E (z + L) = C E (z) .$

The complex constant C be written as

(9.105)

$C = exp (- j β L),$

where β is the complex propagation constant of the periodic media. Thus, one can write

(9.106)

$E (z) = E_{β} (z) exp (- j β z) .$

From Equation 9.106,

(9.107)

$E (z + L) = E_{β} (z + L) exp (- j β (z + L)) .$

Also from Equations 9.104 and 9.106,

(9.108)

$E (z + L) = E (z) exp (- j β L) = E_{β} (z) exp (- j β z) exp (- j β L) = E_{β} (z) exp (- j β (z + L)) .$

From Equations 9.107 and 9.108, it follows that E_β(z) is periodic:

(9.109)

$E_{β} (z + L) = E_{β} (z) .$

Equation 9.106, where E_β(z) is periodic, is called the Bloch wave condition. Such a periodic function can be expanded in a Fourier series:

(9.110)

$E_{β} (z) = \sum_{m = - \infty}^{\infty} A_{m} exp (- j 2 m π z / L),$

and E(z) can be written as

(9.111)

$E (z) = \sum_{m = - \infty}^{\infty} A_{m} exp [- j (β + 2 m π / L) z] = \sum_{m = - \infty}^{\infty} A_{m} exp (- j β_{m} z),$

where

(9.112)

$β_{m} = β + \frac{2 m π}{L} .$

Taking into account both positive-going and negative-going waves, E(z) can be expressed as [1]

(9.113)

$E (z) = \sum_{m = - \infty}^{\infty} A_{m} exp (- j β_{m} z) + \sum_{m = - \infty}^{\infty} B_{m} exp (+ j β_{m} z) .$

Let us next consider the wave propagation in a periodic layered media with plasma layers alternating with the free space. The geometry of the problem is shown in Figure 9.11. Adopting the notation of [12] a unit cell consists of free space from −l < z < l and a plasma layer of plasma frequency ω_p0 from l < z < l + d. The thickness of the unit cell is L = 2l + d. The electric field E(z) in the two layers of the unit cell can be written as

(9.114)

$E (z) = A exp (- j \frac{ω}{c} z) + B exp (j \frac{ω}{c} z), - l \leq z \leq l,$

(9.115)

$E (z) = C exp [- j n \frac{ω}{c} (z - l)] + D exp [j n \frac{ω}{c} (z - l)], l \leq z \leq l + d,$

where n is the refractive index of the plasma medium:

(9.116)

$n = \sqrt{ε_{p}} = \sqrt{1 - ω_{p 0}^{2} / ω^{2}} .$

FIGURE 9.11
Unit cell of unbounded periodic media consisting of layer of plasma.

The continuity of tangential electric and magnetic fields at the boundary translates into the continuity of E and ∂E/∂z at the interfaces:

(9.117)

$E (l^{-}) = E (l^{+}),$

(9.118)

$\frac{\partial E}{\partial z} (l^{-}) = \frac{\partial E}{\partial z} (l^{+}),$

(9.119)

$E (l + d^{-}) = E (l + d^{+}),$

(9.120)

$\frac{\partial E}{\partial z} (l + d^{-}) = \frac{\partial E}{\partial z} (l + d^{+}) .$

From Equations 9.117 and 9.118,

(9.121)

$A exp (- j \frac{ω}{c} l) + B exp (j \frac{ω}{c} l) = C + D,$

(9.122)

$A exp (- j \frac{ω}{c} l) - B exp (j \frac{ω}{c} l) = n (C - D) .$

Since l + d⁺ = −l⁺ + L, from Equation 9.104,

(9.123)

$E (l + d^{+}) = E (- l^{+} + L) = exp (- j β L) E (- l^{+}) .$

From Equations 9.114, 9.115, 9.119, and 9.123

(9.124)

$C exp (- j n \frac{ω}{c} d) + D exp (j n \frac{ω}{c} d) = exp (- j β L) [A exp (- j \frac{ω}{c} l) + B exp (j \frac{ω}{c} l)] .$

From Equations 9.114, 9.115, 9.120, and 9.123

(9.125)

$\begin{array}{l} - n \frac{ω}{c} C exp (- j n \frac{ω}{c} d) + n \frac{ω}{c} D exp (j n \frac{ω}{c} d) \\ = exp (- j β L) [- j \frac{ω}{c} A exp (- \frac{j ω l}{c}) + j \frac{ω}{c} B exp (j \frac{ω}{c} l)] . \end{array}$

Equations 9.121, 9.122, 9.124, and 9.125 can be arranged as a matrix as

(9.126)

$[\begin{matrix} e^{- j ω l / c} & e^{+ j ω l / c} & - 1 & - 1 \\ e^{- j ω l / c} & - e^{j ω l / c} & - n & n \\ e^{- j ω l / c} & e^{+ j ω l / c} & - e^{j (β L - n ω d / c)} & - e^{j (β L + n ω d / c)} \\ e^{- j ω l / c} & - e^{+ j ω l / c} & - n e^{j (β L - n ω d / c)} & n e^{j (β L + n ω d / c)} \end{matrix}] [\begin{matrix} A \\ B \\ C \\ D \end{matrix}] = 0.$

A nonzero solution for the fields can be obtained by equating the determinant of the square matrix in Equation 9.126 to zero. This leads to the dispersion relation

(9.127)

$cos β L = cos \frac{n ω d}{c} cos \frac{2 ω l}{c} - \frac{1}{2} [n + \frac{1}{n}] sin \frac{n ω d}{c} sin \frac{2 ω l}{c} .$

Section 7.4 gives an alternative way of obtaining the dispersion relation based on transmission line analogy and the ABCD parameters.

The above equation can be studied by using reference values β_r and ω_r = β_rc = 2πc/λ_r. Thus, Equation 9.127, in the normalized form, can be written as

(9.128)

$cos (\frac{β}{β_{r}} \frac{2 π L}{λ_{r}}) = cos (\frac{n ω}{ω_{r}} \frac{2 π d}{λ_{r}}) cos (\frac{2 ω}{ω_{r}} \frac{2 π l}{λ_{r}}) - \frac{1}{2} (n + \frac{1}{n}) sin (\frac{n ω}{ω_{r}} \frac{2 π d}{λ_{r}}) sin (\frac{2 ω}{ω_{r}} \frac{2 π l}{λ_{r}}) .$

Figure 9.12 shows the graph ω/ω_r versus β/β_r for the following values of the parameters: L = 0.6λ_r, d = 0.2λ_r, and ω_p0 = 1.2ω_r. We note from Figure 9.12 that the wave is evanescent in the frequency band 0 < ω/ω_r < 0.611. This stop band is due to the plasma medium in the layers. If the layers are dielectric, this stop band will not be present. We have again a stop band in the frequency domain 0.877 < ω/ω_r < 1.225. This stop band is due to the periodicity of the medium. Periodic dielectric layers do exhibit such a forbidden band. This principle is used in optics to construct Bragg reflectors with extremely large reflectance [9]. Figure 9.13 shows the ω–β diagram for dielectric layers with the refractive index n = 1.8. The first stop band in this case is given by 0.537 < ω/ω_r < 0.779.

FIGURE 9.12
Dispersion relation of a periodic plasma medium for d = 0.2λ_r and L = 0.6λ_r.

FIGURE 9.13
Dispersion relation of periodic dielectric layers for d = 0.2λ_r and L = 0.6λ_r. The refractive index of the dielectric layer is 1.8.

9.8Surface Waves

Let us backtrack a little bit and consider the oblique incidence of a p wave on a plasma half-space. In this case, only the outgoing wave will be excited in the plasma half-space and Equation 9.75 becomes

(9.129)

$[\begin{array}{c} 1 & - 1 \\ C η_{y 1} & 1 \end{array}] [\begin{array}{c} E_{x 1}^{P} \\ E_{x}^{R} \end{array}] = [\begin{array}{c} 1 \\ 1 \end{array}] E_{x}^{I} .$

Solution of Equation 9.129 gives the fields of the reflected wave and the wave in the plasma half-space in terms of the fields of the incident wave. If $E_{x}^{I}$ is zero (no incident wave), we expect $E_{x 1}^{P} = E_{x}^{R} = 0$ . This is true in general with one exception. The exception occurs when the determinant of the square matrix on the LHS of Equation 9.129 is zero. In such a case, $E_{x 1}^{P}$ and $E_{x}^{R}$ can be nonzero even if $E_{x}^{I}$ is zero, indicating the possibility of the existence of fields in free space even if there is no incident field. For the exceptional solution, the reflection coefficient is infinity, that is, $E_{x}^{R} \neq 0$ , but $E_{x}^{I} = 0$ . The solution is an eigenvalue solution and gives the dispersion relation for surface plasmons. From Equations 9.73 and 9.129, the dispersion relation is obtained as

(9.130)

$1 + C η_{y 1} = 1 + C \frac{ε_{p}}{q_{1}} = 0.$

Noting that

(9.131)

$k_{0} q_{1} = \sqrt{k_{0}^{2} ε_{p} - k_{x}^{2}}$

and

(9.132)

$k_{0}^{2} C^{2} = k_{0}^{2} - k_{x}^{2},$

where

(9.133)

$k_{0} = \frac{ω}{c},$

(9.134)

$k_{x} = k_{0} S,$

the dispersion relation can be written as

(9.135)

$k_{x}^{2} = \frac{ω^{2}}{c^{2}} \frac{ε_{p}}{1 + ε_{p}} .$

The exponential factors for z < 0 and z > 0 can be written as

(9.136)

$ψ^{P 1} = exp [- j k_{x} x - j k_{0} q_{1} z], z > 0,$

(9.137)

$ψ^{R} = exp [- j k_{x} x + j k_{0} C z], z < 0.$

It can be shown that k_x will be real and both

(9.138)

$α_{1} = j k_{0} q_{1}$

and

(9.139)

$α_{2} = j k_{0} C = j \sqrt{k_{0}^{2} - k_{x}^{2}}$

will be real and positive if

(9.140)

$ε_{p} < - 1.$

When Equation 9.140 is satisfied

(9.141)

$ψ^{P 1} = exp (- α_{1} z) exp (- j k_{x} x), z > 0,$

(9.142)

$ψ^{R} = exp (α_{2} z) exp (- j k_{x} x), z < 0,$

where α₁ and α₂ are positive real quantities. Equations 9.141 and 9.142 show that the waves, while propagating along the surface, attenuate in the direction normal to the surface. For this reason, the wave is called a surface wave. In a plasma when $ω < ω_{p} / \sqrt{2},$ ε_p will be less than −1. Thus, an interface between free space and a plasma medium can support a surface wave.

Defining the refractive index of the surface mode as n, that is,

(9.143)

$n = \frac{c}{ω} k_{x},$

Equation 9.135 can be written as

(9.144)

$n^{2} = \frac{Ω^{2} - 1}{2 (Ω^{2} - 1 / 2)}, Ω < \frac{1}{\sqrt{2}} .$

For $0 < ω < ω_{p} / \sqrt{2},$ n is real and the surface wave propagates. In this interval of ω, the dielectric constant ε_p < −1. The surface wave mode is referred to, in the literature, as Fano mode [13] and also as a nonradiative surface plasmon. In the interval 1 < Ω < ∞, n² is positive and less than 0.5. However, the α-values obtained from Equations 9.138 and 9.139 are imaginary. The wave is not bound to the interface and the mode, called Brewster mode, in this case is radiative and referred to as a radiative surface plasmon. Figure 9.14 sketches ε_p(ω) and n² (ω). Another way of presenting the information is through the Ω−K diagram, where Ω and K are normalized frequency and wave number, respectively, that is,

(9.145)

$Ω = \frac{ω}{ω_{p}},$

(9.146)

$K = \frac{c k_{x}}{ω_{p}} = n Ω .$

FIGURE 9.14
Refractive index for surface plasmon modes.

From Equations 9.143 and 9.144, we obtain the relation

(9.147)

$K^{2} = n^{2} Ω^{2} = \frac{Ω^{2} (Ω^{2} - 1)}{2 Ω^{2} - 1} .$

Figure 9.15 shows the Ω–K diagram.

FIGURE 9.15
Ω−K diagram for surface plasmon modes.

The Fano mode is a surface wave called the surface plasmon and exists for $ω < ω_{p} / \sqrt{2} .$ The surface mode does not have a low-frequency cutoff. It is guided by an open-boundary structure. It has a phase velocity less than that of light. The eigenvalue spectrum is continuous. This is in contrast to the eigenvalue spectrum of a conducting-boundary waveguide that has an infinite number of discrete modes of propagation at a given frequency.

The application of photons (electromagnetic light waves) to excite the surface plasmons meets with the difficulty that the dispersion relation of the Fano mode lies to the right of the light line Ω = K (see Figure 9.15). At a given photon energy ℏω, the photon momentum ℏk_x has to be increased by ℏΔk_x = ℏ(ω_p/c)ΔK in order to transform the photons into surface plasmons. The two techniques commonly used to excite surface plasmons are (a) grating coupler (b) attenuated total reflection (ATR) method. A complete account of surface plasmons can be found in [13,14].

Surface waves can also exist at the interface of a dielectric with a plasma layer or a plasma cylinder. The theory of surface waves on a gas-discharge plasma is given in [15]. Kalluri [16] used this theory to explore the possibility of a backscatter from a plasma plume see Appendix 9A for details. Moissan et al. [17] achieved plasma generation through surface plasmons in a device called Surfatron. It is a highly efficient device for launching surface plasmons that produce plasma at a microwave frequency.

9.9Transient Response of a Plasma Half-Space

Reflection of transient pulses from a sharply bounded plasma half-space can be obtained by finding the response of the plasma to a time-harmonic wave. For example, if R(jω) is the reflection coefficient, then the response to an impulsive plane wave e_r (t) can be obtained [18].

Let e₀(t) = δ(t) be the electric field of the impulsive plane wave at z = 0. Its Laplace transform is

(9.148)

$E_{0} (s) = L [e_{0} (t)] = L [δ (t)] = 1.$

If the reflected field is designated at the point of incidence on the interface e_r (t), then its Laplace transform E_r(s) is given by

(9.149)

$E_{r} (s) = R (s) E_{0} (s) = R (s),$

where R(s) is obtained by replacing ( jω) by s in R(jω): the Laplace inverse of R(s) gives e_r(t):

(9.150)

$e_{r} (t) = L^{- 1} R (s) = \frac{1}{2 π j} \int_{σ_{1} - j \infty}^{σ_{1} + j \infty} R (s) e^{s t} d s .$

A direct analytical evaluation of Equation 9.150 is modified to suit the numerical evaluation [19]

(9.151)

$e_{r} (t) = \frac{2}{π} \int_{0}^{\infty} Re [R (j ω)] cos ω t d ω,$

where is assumed e_r(t) = 0 for t < 0 and Re stands for real.

9.9.1Isotropic Plasma Half-Space s Wave

Consider oblique incidence of an s wave. It can be shown [18] that

(9.152)

$R_{s} (j ω) = \frac{j ω C - j^{2} ω^{2} C^{2} + ω_{p}^{2}}{j ω C + j^{2} ω^{2} C^{2} + ω_{p}^{2}},$

(9.153a)

$R_{s} (s) = \frac{{[s - {(s^{2} + ω_{p 1}^{2})}^{1 / 2}]}^{2}}{ω_{p 1}^{2}},$

where

(9.153b)

$ω_{p 1} = \frac{ω_{p}}{C} .$

(9.153c)

$C = cos θ,$

and θ is the angle of incidence. From a standard Laplace transform table [11],

(9.154)

$L^{- 1} {[{(s^{2} + b^{2})}^{1 / 2} - s]}^{ν} = \frac{υ b^{ν}}{t} J_{ν} (b t), ν > 0.$

Thus, one obtains the Laplace inverse of Equation 9.153, leading to

(9.155)

$ω_{p}^{- 1} e_{r} (t) = - \frac{2}{ω_{p} t} J_{2} (\frac{ω_{p} t}{C}) .$

Figure 9.16 shows the impulse response for various angles of incidence. The effect of dispersion is clearly visible. There is an appreciable effect of the angle of incidence on the amplitude and the period of oscillation of the response. It is seen that as the angle of incidence decreases (or C increases), the amplitude of the response (especially the early part) decreases, whereas the period of the oscillation increases. The variation of Re[R(Ω)] with Ω = ω/ω_p is shown as an inset for C = 0.4.

FIGURE 9.16
Impulse response of isotropic plasma half-space to an s-wave incidence.

9.9.2Impulse Response of Several Other Cases Including Plasma Slab

References [18,19,20,21,22,23,24,25,26] and the references in these citations give access to the additional literature on this topic.

9.10Solitons [27]

This topic deals with a desirable aspect of the EMW transformation by the balancing act of two complexities each of which by itself produces an undesirable effect. The balancing of the wave-breaking feature of a nonlinear medium complexity with the flattening feature of the dispersion leads to a soliton solution. Hirose and Longren [27] give a transmission line analogy to explain the balancing of the two dominant effects of the complexities of dispersion and nonlinearity. Ricketts and Ham [28] discuss the theory, design, and application of electrical solitons.

9.11Perfect Dispersive Medium

As we have seen in Figure 9.16, a dispersive medium distorts a pulse by modulating the instantaneous frequency contents of the pulse. Gupta and Caloz [29] have explained the concept of a perfect dispersive medium with a perfectly flat transmission coefficient magnitude over the entire spectrum leading to an improved real-time signal processing technology. It is an engineered medium of “stacked loss-gain metasurfaces” [29].

Several such new ideas are being pursued under the broad banner of dispersion-engineered space-time modulated systems.

References

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Table of Contents for 9. Waves in Isotropic Cold Plasma: Dispersive Medium

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9.4Reflection by a Plasma Slab [5]

9.5Tunneling of Power through a Plasma Slab [5, 7]

9.10Solitons [27]

Table of Contents for
9. Waves in Isotropic Cold Plasma: Dispersive Medium