10A.2.1Waves Outside the Slab

For the incident, reflected, and transmitted (I, R, and T) waves outside the plasma slab, the electric and magnetic fields obliquely incident can be described as

where ${\overrightarrow{E}}_0^{\text{I},\text{R},\text{T}}$ and ${\overrightarrow{H}}_0^{\text{I},\text{R},\text{T}}$ are the complex electric and magnetic amplitudes given by

ψ^I,R,T is the phase reference of the waveform at $\overrightarrow{r}=0$,

From Figure 10A.1, the geometry of the wave vector can be described as

where S^T = k_I/k_TS and $C^{\text{T}}=\sqrt{1-{\left(k_{\text{I}}\text{}/\text{}{k}_{\text{T}}S\right)}^2}$. S^T and C^T describe the characteristics of the dielectric in the output of the system. Note, S = sin(θ^I), C = cos(θ^I), S^T = sin(θ^T), and C^T = cos(θ^T). If k_I = k_T, then S^T = S and C^T = C. Since the position vector is $\overrightarrow{r}=\widehat{x}x+\widehat{y}y+\widehat{z}z$, the phase factor of the waves outside the plasma slab become

where $C^{\prime }=\sqrt{{\left(k_{\text{T}}\text{}/\text{}{k}_{\text{I}}\right)}^2-S^2}=\left(k_{\text{T}}{k}_{\text{I}}\right)C^{\text{T}}$. From ψ^T, $k_{\text{T},x}^2+k_{\text{T},z}^2=k_{\text{T}}^2$, where $k_{\text{T}}\mathrm{\omega }\text{}/\text{}c\sqrt{{\mathrm{\epsilon }}_{\text{T}}}=\mathrm{\omega }\text{}/\text{}c{n}_{\text{T}}=k_0{n}_{\text{T}}$. The angle-dependent parameter, C′, in ψ^T (Equation 10A.5) was derived using Snell’s law. Snell’s law implies that the boundary conditions (or phase match) has been satisfied as is evident in the x-component of (Equation 10A.5) and also by the fact that θ^I = θ^R. The x-components of the phase factor for the incident, reflected, and transmitted waves are the same to satisfy the boundary conditions, and hence, for the TM wave the tangential components of $\overrightarrow{E}$ and $\overrightarrow{H}$ are continuous across the boundary in both space and time.

For a plane wave propagation [9], the electric and magnetic fields can be described as

where μ₀ is the free-space permeability. Using Equation 10A.1 in Equation 10A.6, we obtain

where η^I = η₀/n_I.

10A.2.2Waves Inside the Slab

The basic equations for a warm plasma are Maxwell’s equations:

Using the hydrodynamic approximation for a compressible electron gas the conservation of momentum [5] (isotropic case)

The conservation of energy is

The equation of state is

where N, N₀ are the small and large signal number density (m⁻³), p, p₀ the small and large signal pressure (kg/m s²), T, T₀ the small and large signal temperature (K), °K the Boltzman constant, γ the specific heat in constant pressure/specific heat in constant volume, m the electron mass (kg), $\overrightarrow{u}$ the velocity vector (m/s).

The formulation given above is based on the linear theory. Unz showed that Equations 10A.8 and 10A.9 reduce to the wave equation

where

and Z is described in terms of the electron collision frequency ν and transmission frequency ω:

The term

is inversely proportional to the normalized frequency squared, X = 1/Ω², where Ω = ω/ω_p. Within the film four modes exist: Transmitted/reflected transverse waves and transmitted/reflected longitudinal waves. The boundary at z = 0, d serves to act as the source of an acoustic wave mode only supported within the slab medium. Within the slab, a TM wave and an acoustic wave with velocity, pressure, and electric field intensity, both satisfy the wave equation (10A.10). In this chapter, the dimensionless parameter n_p can be described in terms of the input medium, input angle, and plasma characteristic root parameter, q′:

where q′ will be determined. Note: q′ is not the derivative of q but rather distinguishes between a n_I > 1 dependent solution of the dispersion equation and a n_I = 1 solution. For a warm plasma, the velocity of the longitudinal component is significant and in certain metals is approximately two to three orders of magnitude less of the speed of light c. The electric and magnetic field intensities inside the plasma can be described as

The electric and magnetic fields in Equation 10A.15 have complex field amplitudes as

The phase of the fields within the plasma slab is defined as

When using Equations 10A.15 and 10A.16 in Equation 10A.10, one has for oblique incidence on the slab interface:

Equation 10A.18a, Equation 10A.18b and Equation 10A.18c can be rearranged in a matrix form as

which is

The components of Equation 10A.20 come from Equation 10A.18a, Equation 10A.18b and Equation 10A.18c and are as follows:

Setting determinant of [F] to zero yields the dispersion relation F₁₁F₃₃ − F₁₃F₃₁ = 0. The characteristic roots for the cold and warm plasma models are derived directly from the dispersion relation

Subscript 1 indicates an upward-going wave and subscript 2 indicates a reflected downward-going wave of the transverse type, similarly the subscripts 3 and 4 of the longitudinal type. Using Equation 10A.8 to find the magnetic field intensity yields

Since the incident wave is a TM wave, only the y-component matters, and hence

From dispersion relation and Equation 10A.24, the ratio of the magnetic to electric tangential components becomes

The warm plasma electric field-dependent electron velocity field is given by

where β₀ = eN₀/jωε₀ = ωmX/je. Using Equations 10A.21a and 10A.21c, Equation 10A.26 becomes

10A.2.3Power Reflection Coefficient for a Warm Isotropic Plasma Slab between Two Different Dielectric Media (i.e., n_I and n_T)

In considering the case where the plasma medium is warm, a longitudinal electric field must be taken into account. The power reflection coefficient is derived for a warm plasma sandwiched between two different dielectric media n_I and n_T. Using Equations 10A.25 and 10A.27 with the following equations for a collisionless plasma, U = 1, we have

We use boundary conditions where the tangential components of $\overrightarrow{E}$ and $\overrightarrow{H}$ are continuous at z = 0 and d to generate four equations. For time-harmonic waves, tangential boundary conditions are used as they are derived by Maxwell’s curl equations. Normal boundary conditions (from divergence equations) can be derived from the curl equations. Hence, normal boundary conditions are not independent but are assured to be satisfied when the tangential boundary conditions are satisfied [10]. Two more equations are generated from the boundary conditions to account for the z component of the electron velocity to be zero at the interfaces since there is no electron flow within the dielectrics. Writing the system of equations from the boundary conditions, Equation 10A.1, Equation 10A.2, Equation 10A.3, Equation 10A.4, Equation 10A.5, Equation 10A.6 and Equation 10A.7 and (10A.28):

The longitudinal component of the waveform inside the plasma is comprised of the electric field only, since η_y3 is equal to zero. Putting Equation 10A.29 in a matrix form gives

where ${\mathrm{\lambda }}_i^{\prime }={\text{e}}^{-\text{j}{k}_0{n}_{\text{I}}{q}_i^{\prime }d},E_x^{{\mathrm{T}}^{\prime }}=E_x^{\text{T}}{\text{e}}^{-\text{j}{k}_0{n}_{\text{I}}{C}^{\prime }d},\text{and}{H}_y^{{\mathrm{T}}^{\prime }}=H_y^{\text{T}}{\text{e}}^{-\text{j}{k}_0{n}_{\text{I}}{C}^{\prime }d}$.

The complex reflection coefficient can be solved by the ratio of the reflected and incident electric fields

The symbol ‖ implies that the incident and reflected waves are p-waves on the plane of incidence. After substantial simplification of Equation 10A.31, _‖R_‖ becomes

where

The power reflection coefficient is defined by the magnitude squared of _‖R_‖

A closed form of Equation 10A.34 is not given here because of its high complexity; it can either be studied using computer code or from approximation. The refractive indices n_I and n_T cause the angle-dependent parameter C′ to become zero at the critical angle. The real and imaginary parts of the complex reflection coefficient change depending on whether the incident angle is less than or greater than the critical angle. Equation 10A.34 is much dependent on n_I and n_T such that the choice of material will dramatically change the reflection properties of the composite material. Figure 10A.2 shows the power reflection coefficient for a normalized film thickness of d_p = d/λ_p = 0.2, where λ_p is the plasma wavelength, λ_p = 2πc/ω_p, an input angle of θ^I = 60° and materials n_I = 1.1 and n_T = 1.3 with δ = 0.1. The power reflection coefficient changes from a monotonic to a complex oscillatory behavior that occurs between zero and one with a periodicity that decreases with increasing frequency.

10A.3.1Power Reflection Coefficient for Region 1, 0 < Ω < [1 − (δn_IS)²]^−1/2

Based on Figure 10A.3, the respective characteristic roots are

All of the ${\mathrm{\lambda }}_i^{\prime }$ terms will be real, and hence, Equation 10A.32 will consist of hyperbolic sines and hyperbolic cosines only. As a result, there is no propagation of either transverse or longitudinal modes and the waves are evanescent. The power reflection coefficient will have no oscillations and will stay at unity for this region. The following section will show that as the film gets sufficiently thin, power tunneling will take effect and ρ_‖ will not always be unity, but will allow some power to propagate, or tunnel, through the slab.

10A.3.1.1Region 1 and the Effects of Power Tunneling

For thick slabs, the waves are evanescent and the decaying fields will not interact. When the slab becomes sufficiently thin power tunneling will occur. The tunneling represents itself as a peak in region 1 of the transmission coefficient τ_‖. Kalluri [11] gave an expression at the peak which is a transcendental equation related to the phase term of oscillation in ρ_‖. Tuning Ω and measuring the peak in τ_‖ for a known incident angle gives Ω_max that can lead to determining the slab thickness or the plasma frequency.

Figure 10A.4 shows the effect of power tunneling for normalized thicknesses of d_p = 0.05, 0.1, 0.2, 1, and n_I = n_T = 1. The power transmission coefficient, τ_‖ = 1 − ρ_‖, is also shown. For thin films at λ_p/20, the slab is almost completely transmissive. For d_P = 1, the intensity of light impinging on the warm plasma slab is completely reflected away from the interface. However, as the thickness of the slab decreases, the cross-interaction between evanescent forward-going electric and backward-going magnetic (and vice versa) fields create a real propagating power in a plane parallel to the interface that emerges at z = d. This explains the dip seen in this first region. For the air–plasma–air cold plasma case, an experimental measurement of the maximum power transfer could yield information about the material thickness with the use of a transcendental equation. As the thickness of the slab increases, this phenomenon decreases as the decaying fields within the slab become negligibly small as they reach the opposite interface. Figure 10A.5 shows results for dielectric–plasma–dielectric cold and warm plasma case, n_I = 1.46, n_T = 1.48. For all thicknesses, there is a noticeable difference in the tunneling effect between the cold plasma model (δ = 0) and the warm plasma case (δ = 0.1) and there are differences when different dielectrics are introduced.

Looking at both Figures 10A.4 and 10A.5, the largest difference between warm and cold is seen in ρ_‖ at d_p = 0.05 near the plasma frequency. For all thicknesses, more light transmits through the slab for the cold plasma model. In the measurement of optical constants, the dip seen for 0 ≤ Ω ≤ 1 can provide valuable information about the material n_p with knowledge of n_I and n_T by an appropriate correction factor to the transcendental equation. In addition, the figures show that at Ω = 0, 1, ρ_‖ = 1. From Equation 10A.32 as Ω → 0 (ω = 0), X → ∞ then as shown in Figure 10A.3, q₁ → ∞ and q₃ → ∞. Subsequently, η_y1, β_z1 → 0, β_z3, ${\mathrm{\lambda }}_i^{\prime }\to \infty$ and _‖R_‖ → 1. When Ω = 1 (ω = ω_p), then X → 1, η_y1 → 0, β_z1, β_z3 → −1 and ${\mathrm{\lambda }}_i^{\prime }$ will all be equal, then _‖R_‖ → 1. For both Ω = 0, 1, ρ_‖ is forced to unity, independent of any n_I or n_T creating a confluence point for any choice of material.

10A.3.1.2Region 1 and Surface Plasmons

If one considers the electron plasma wave velocity a to become several orders of magnitudes slower than c = 3 × 10⁸ m/s, then from Equations 10A.22 and 10A.28b, δ → 0, q₃ → ∞, β_z3 → ∞ and the plasma is cold. Also, from Equation 10A.32, only the terms multiplied by ${\mathrm{\beta }}_{z3}^2$ survive and one can derive the power reflection coefficient for a cold plasma between two different dielectrics,

(10A.37)$${\mathrm{\rho }}_{\parallel }^2=\left|\frac{\left(1-C{\mathrm{\eta }}_{y1}^{\prime }\mathrm{/}{n}_{\mathrm{I}}^2\right)\left(1-C^{\prime }{\mathrm{\eta }}_{y2}^{\prime }\mathrm{/}{n}_{\mathrm{I}}^2\left({C^{\prime }}^2+S^2\right)\right){\mathrm{\lambda }}_2^{\prime }-\left(1-C{\mathrm{\eta }}_{y2}^{\prime }\mathrm{/}{n}_{\mathrm{I}}^2\right)\left(1-C^{\prime }{\mathrm{\eta }}_{y1}^{\prime }\mathrm{/}{n}_{\mathrm{I}}^2\left({C^{\prime }}^2+S^2\right)\right){\mathrm{\lambda }}_1^{\prime }}{\left(1+C{\mathrm{\eta }}_{y1}^{\prime }\mathrm{/}{n}_{\mathrm{I}}^2\right)\left(1-C^{\prime }{\mathrm{\eta }}_{y2}^{\prime }\mathrm{/}{n}_{\mathrm{I}}^2\left({C^{\prime }}^2+S^2\right)\right){\mathrm{\lambda }}_2^{\prime }-\left(1+C{\mathrm{\eta }}_{y2}^{\prime }\mathrm{/}{n}_{\mathrm{I}}^2\right)\left(1-C^{\prime }{\mathrm{\eta }}_{y1}^{\prime }\mathrm{/}{n}_{\mathrm{I}}^2\left({C^{\prime }}^2+S^2\right)\right){\mathrm{\lambda }}_1^{\prime }}\right|.$$

Plotting Equation 10A.37, Figure 10A.6, using the material chosen by Lekner [12] and Otto [13] the input dielectric was glass of n_I = 1.9018, silver film of n_p = 0.055 + j3.28, which is based on the plasma frequency, and lithium fluoride of n_T = 1.392 as the output medium. Collisions were taken into account as is indicated by the real part of n_p.

The result in Figure 10A.6 compares quite well with Lekner’s result. From the results of Equation 10A.37, the collision term of n_p causes a nonzero real part of q′. Since this is the case of ATR, the frequency corresponding to the wavelength of λ = 546.1 nm is below the plasma frequency and in the region of total reflection. Figure 10A.6 also shows the C′ term, which can be shown to be zero at the critical angle $C^{\prime }=\sqrt{{\left(n_{\text{T}}\text{/}{n}_{\text{I}}\right)}^2-S^2\left(\mathrm{\theta }=47.05°\right)}=0$ causing ρ_‖ to transition downward to zero as photons couple to electrons exciting a surface plasmon at the surface plasmon angle of 53.9°. The critical angle (47.05° for this case) is the point where C′ turns from a purely real number to a purely imaginary number. This zero point is the angle at which total internal reflection would occur and the wave would travel along the boundary if no film was present between n_I and n_T. The critical angle is also the point where the analytical expressions for the real and imaginary parts of _‖R_‖ (not shown in this chapter) change, hence, $\text{Re}{\left\{{}_{\parallel }{R}_{\parallel }\right\}}_{{\mathrm{\theta }}^{\text{I}}<{\mathrm{\theta }}^C}\ne \text{Re}{\left\{{}_{\parallel }{R}_{\parallel }\right\}}_{{\mathrm{\theta }}^{\text{I}}>{\mathrm{\theta }}^C}$ and $\text{Im}{\left\{{}_{\parallel }{R}_{\parallel }\right\}}_{{\mathrm{\theta }}^{\text{I}}<{\mathrm{\theta }}^C}\ne \text{Im}{\left\{{}_{\parallel }{R}_{\parallel }\right\}}_{{\mathrm{\theta }}^{\text{I}}>{\mathrm{\theta }}^C}$. The surface plasmon resonance is the effect of a metal film present sandwiched between n_I and n_T. In a sense, the insertion of the metal film induces an equivalent Brewster angle for an incident p-wave that would not exist for the half-space materials only. From Averitt [14], the angle where the power reflection coefficient is totally suppressed, and for the real part of ε^P the surface plasmon resonance can be calculated as

(10A.38)$${\mathrm{\theta }}_{\text{SP}}=arcsin\left(\frac{\sqrt{{\mathrm{\epsilon }}^{\text{T}}{\mathrm{\epsilon }}^{\text{P}}\text{/}\left({\mathrm{\epsilon }}^{\text{T}}+{\mathrm{\epsilon }}^{\text{P}}\right)}}{\sqrt{{\mathrm{\epsilon }}^{\text{I}}}}\right)=53.9°.$$

10A.3.2Power Reflection Coefficient for Region 2, [1 − (δn_IS)²]^−1/2 Ω < [1 − (n_IS)²]^−1/2

Based on Figure 10A.3, the respective characteristic roots are

The power reflection coefficient takes on an oscillatory dynamic. The purely real q₃ implies that the longitudinal (acoustic) mode will propagate through the film. The ${\mathrm{\lambda }}_i^{\prime }$ terms in Equation 10A.32 containing q₃ will be complex implying propagation of power through the slab. As can be seen in Figure 10A.2, the oscillations in this region are dramatic as a result of the spatially dispersive plasma medium. The warm plasma component is an acoustic mode of electrostatic field intensity with amplitude and velocity a, it is longitudinal, and is a solution to Maxwell’s equations. When a is small compared to c, there is no wave propagation and the plasma becomes cold, then only plasma oscillations exist at ω_p. The ${\mathrm{\lambda }}_i^{\prime }$ terms containing q₁ will be real and the transverse modes will be evanescent.

10A.3.2.1Region 2 and an Approximation of ρ_‖ for n_I = n_T = 1

Since, sin(k₀jq₁d) ≡ j sinh(k₀q₁d) and cos(k₀jq₁d) ≡ cosh(k₀q₁d),

(10A.40)$${\mathrm{\rho }}_{\parallel }=\frac{1}{1+{\left[\begin{array}{c}2C{q}_3\left(1-X\right)\left[q_1{q}_3\sin \left(k_0{q}_{3}d\right)+S^{2}X\sinh \left(k_0{q}_{1}d\right)\right]/\hfill \\ 2q_1{q}_3{S}^{2}X\left[1-\cosh \left(k_0{q}_{1}d\right)\cos \left(k_0{q}_{3}d\right)+\left[-q_1^2{q}_3^2-C^2{q}_3^2{\left(1-X\right)}^2+S^4{X}^2\right]\sinh \left(k_0{q}_{1}d\right)\sin \left(k_0{q}_{3}d\right)\right]\hfill \end{array}\right]}^2}.$$

The power reflection coefficient in this region can be approximated and put into a simpler form and can be written as

(10A.41)$${\mathrm{\rho }}_{\parallel }=\frac{1}{1+{\left[A_0\left[q_1{q}_{3}sin\left(k_0{q}_{3}d\right)+S^{2}Xsinh\left(k_0{q}_{1}d\right)\right]\text{/}\left(A_2+A_{3}cos\left(k_0{q}_{3}d\right)+A_{4}sin\left(k_0{q}_{3}d\right)\right)\right]}^2},$$

where

(10A.42a)$$A_0=2C{q}_3\left(1-X\right),$$

(10A.42b)$$A_1=\left[-q_1^2{q}_3^2-C^2{q}_3^2{\left(1-X\right)}^2+S^4{X}^2\right],$$

(10A.42c)$$A_2=2q_1{q}_3{S}^{2}X,$$

(10A.42d)$$A_3=-A_{2}cosh\left(k_0{q}_{1}d\right),$$

(10A.42e)$$A_4=A_{1}sinh\left(k_0{q}_{1}d\right).$$

The A₂ term does not contribute much to Equation 10A.41 when compared to A₃ and A₄. The A₃ and A₄ terms are large and negative for most of the region of interest, as such, A₂ can be neglected. An approximation to Equation 10A.41 (designated as ρ_a) becomes

(10A.43)$${\mathrm{\rho }}_{\text{a}}=\frac{1}{1+{\left[A_0\left[q_1{q}_{3}sin\left(k_0{q}_{3}d\right)+S^{2}Xsinh\left(k_0{q}_{1}d\right)\right]\text{/}\left(A_{3}cos\left(k_0{q}_{3}d\right)+A_{4}sin\left(k_0{q}_{3}d\right)\right)\right]}^2},$$

which can be reduced to

(10A.44)$${\mathrm{\rho }}_{\text{a}}=\frac{1}{\begin{array}{c}1+{\left[A_0\left[q_1{q}_{3}sin\left(k_0{q}_{3}d\right)+S^{2}Xsinh\left(k_0{q}_{1}d\right)\right]\text{/}\sqrt{A_3^2+A_4^2}\right]}^2\hfill \\ \times {csc}^2\left(k_0{q}_{3}d+{tan}^{-1}\left(-A_4\text{}/\text{}{A}_3\right)\pm \mathrm{\pi }/2\right)\hfill \end{array}}.$$

Equation 10A.44 is in the same form as the cold plasma slab result reported by Kalluri and Prasad [7]. The plots for ρ_‖ and ρ_a are given in Figure 10A.7 for δ = 0.3 with film thickness of d_p = 1. Region 2 is approximately {1.04 < Ω < 2}. The approximation of ρ_‖ is very good until Ω ≈ 1.86. Values of Ω in the region ~1.86 < Ω < 2 is where A₂ ≈ A₃ ≈ A₄ and the suppression of the A₂ term is no longer valid in Equation 10A.43. The oscillations have a periodicity that decreases as Ω increases, which is caused by the Ω-dependent csc² term. Although there is an oscillatory term in the bracketed term [sin( )]², the zeros of ρ_a are not determined by this term but are completely dependent on the phase of the csc² term, and hence

(10A.45)$$k_0{q}_{3}d+{tan}^{-1}\left(-\frac{A_4}{A_3}\right)\pm \frac{\mathrm{\pi }}{2}=m\mathrm{\pi },m=0,1,2,\dots .$$

At m = 0,1,2, …, the csc² term goes to infinity defining discrete zeros for ρ_a and consequently for ρ_‖. The peaks seen in Figure 10A.7 are at values of Ω where dρ_a/dΩ = 0. If one allows Equation 10A.44 to become

(10A.46)$${\mathrm{\rho }}_{\text{a}}=\frac{1}{1+A_5^2\text{/}\left(A_3^2+A_4^2\right){csc}^2\left(A_6\right)},$$

where A₅ = A₀(q₁q₃ sin(k₀q₃d) + S²X sinh(k₀q₁d)) and A₆ = k₀q₃d + tan⁻¹(−A₄/A₃) ± π/2.

Then from Equation 10A.46, the equation to find the peaks of ρ_a is a transcendental equation,

(10A.47)$$tan\left(A_6\right)=f\frac{\text{d}{A}_6}{\text{d}\mathrm{\Omega }},$$

where

(10A.48)$$f=\frac{\left(A_3^2+A_4^2\right)}{\left(A_3^2+A_4^2\right)\left(1\text{/}\text{}{A}_5\right)\text{d}{A}_5\text{/}\text{}\text{d}\mathrm{\Omega }-\left(A_3\text{d}{A}_3\text{/}\text{}\text{d}\mathrm{\Omega }+A_4\text{d}{A}_4\text{/}\text{}\text{d}\mathrm{\Omega }\right)},$$

and

(10A.49)$$\frac{\text{d}{A}_6}{\text{d}\mathrm{\Omega }}=k_0{q}_{3}d\left(\frac{1}{q_3}\frac{\text{d}{q}_3}{\text{d}\mathrm{\Omega }}+\frac{1}{\mathrm{\Omega }}\right)+\frac{\text{d}}{\text{d}\mathrm{\Omega }}{tan}^{-1}\left(-\frac{A_4}{A_3}\right).$$