The structure of the surface and standing wave resonances and their coupling in the configuration of the overdense plasma slab with a single diffraction grating are studied, using impedance matching techniques. Analytical criteria and exact expressions are obtained for plasma and diffraction grating parameters which define resonance conditions for absolute transparency in the ideal plasma and reflectionless absorption in a plasma with dissipation.

## I. INTRODUCTION

Propagation of electromagnetic waves through overdense plasmas has been of great interest for quite some time due to its practical importance for communications, radars, plasma heating, and other applications. The anomalous (up to 100%) energy transmission through an overdense plasma region of negative permittivity (*ε _{p}* < 0) can be achieved via superposition of decaying and growing evanescent waves (tunneling).

^{1–4}In general, the excitation and amplification of evanescent waves can be supported by coupling to various resonances of inhomogeneous plasma structures which may include both standing wave resonances of propagating waves

^{5}and surface wave (plasmon) resonances of evanescent waves.

^{3}Surface wave (or polariton/plasmon

^{6}) eigen-modes exist at the interface of overdense plasma (

*ε*< 0) and vacuum, or other plasma (dielectric) regions with positive permittivity (

_{p}*ε*> 0).

^{7}Such eigen-modes propagate along the interface and are localized in the perpendicular direction (across the interface). More general types of plasmon modes, including periodic or individual subwavelength structures (such as aperture/slit), can be found in 2D and 3D geometries.

^{8–10}The crucial role of surface wave resonances for phenomena of extraordinary transmission and absorption has been established for a number of models and supported by experiments.

^{2,11,12}Indeed, in some plasma applications, excitation of surface modes has explained the total absorption of electromagnetic waves in overdense plasmas.

^{13}Furthermore, surface mode resonances play an important role in heating of microwave plasmas, see Ref. 14 and references therein. Plasmon resonances also define unique properties of plasma-based metamaterials.

^{15}

An electromagnetic wave propagating in vacuum cannot be matched (coupled) with a surface wave localized at the interface between a vacuum region and a region of negative permittivity. Such coupling, however, can be achieved in configurations of double- or multi-layer structures which include an intermediate plasma (dielectric) region with positive permittivity.^{16,17} This principle serves as a basis for the so called zero-*ε _{eff}* structures

^{3,18,19}with $ \epsilon e f f = \epsilon a a + \epsilon p d = 0 , $ where

*a*is the width of the intermediate layer with $ \epsilon a > 0 $ and

*d*is the width of the plasma layer with

*ε*< 0. In the ideal case without dissipation, such structures have 100% transmissivity for the resonant values of the incidence angle.

_{p}In another approach, the plasmon evanescent surface wave mode can be coupled to the propagating vacuum wave $ \u223c exp \u2009 ( \u2212 i \omega t + i k \xb7 r ) $ via scattering on a periodic subwavelength structure $ \u223c exp \u2009 ( i q \xb7 r ) $. Such scattering will generate sideband evanescent harmonics, $ \xb1 k + q \u2009 , \u2009 $ that can be matched to the surface mode eigen-mode. Periodic spatial modulations can be created via modulation of plasma parameters (e.g., density) with external waves or laser/electron beams.^{10,20} Intrinsic, nonlinear wave generation was also suggested as a possible mechanism.^{15,21–23} Theoretical models for light tunneling via periodically structured metal films have been presented in Refs. 26–30. Alternatively, an external periodic metal diffraction grating can be used,^{24,25} which appears as a more practical solution for plasma applications.^{13,31,32}

In general, the full solution of the transmission and/or reflection problem is obtained by solving a system of coupled linear equations for incident, reflected, and transmitted propagating waves, as well as for standing and evanescent modes inside the plasma layer and the vacuum/dielectric layer structures. Formally, it is a straightforward linear problem, but due to a large number of variables, solving it analytically is technically cumbersome, and the solution is therefore often obtained numerically. As a result, the nature of the involved resonance modes and their coupling is not easily understood. Furthermore, conditions and geometric parameters for optimizing the transmissivity and/or absorption cannot be expressed analytically. Based on the idea of critical coupling in optical waveguides,^{33} a phenomenological model has been proposed to find a quantitative description of resonance coupling and conditions for transmission and absorption in overdense plasmas.^{2,13,32} The plasmon resonator with evanescent eigen-modes has special features, which distinguishes it from standard waveguide type resonators. One important difference is that the energy transport by evanescent modes in tunneling regimes is not described by the group velocity of the wave packets. Instead, one can use the energy flow velocity, which is very different from the group velocity. Typically in tunneling regimes, the energy flow velocity is very small^{34} while the group velocity is not even defined. It is worthwhile to note that the slow energy flow velocity for tunneling structures is the basis for “slow” or “frozen” light concepts and devices. Another feature of the plasmon eigen-modes is that such modes are not really stationary modes in finite size configurations. In configurations with plasma layers of finite widths, the modes are quasistationary, slowly decaying in time, “leaking” energy into the outside region (“leaking” modes).^{35} These are precisely the modes that can be coupled to the propagating mode in free space and amplify the evanescent modes, leading to anomalous transmission and absorption.

In this paper, we present a simple framework for the description of resonance coupling, mode amplification, and transmission in overdense plasma with a single diffraction grating structure. The method we use is based on the standard wave impedance concept and allows for a compact formulation of wave propagation in a wide class of problems with complex multi-layer structures which include overdense plasma layers. It enables us to clarify the nature of resonances and their coupling, and to obtain relatively simple expressions for optimal conditions for wave transmission and absorption in the overdense plasma layer. It allows for easy consecutive calculations of the transmission and reflection coefficients and can be easily generalized for multiple layers and continuous profiles.

## II. BASIC MODEL FOR THE PLASMA LAYER WITH A DIFFRACTION GRATING

We consider the electromagnetic wave incident on the plasma layer preceded by the diffraction grating which is placed at a distance *a* from the plasma boundary as shown in Fig. 1. The transverse magnetic TM or p-polarization is assumed, so that the electromagnetic field has the components **E** = (*E _{x}*,

*E*, 0) and

_{y}**B**=(0, 0,

*B*).

_{z}The magnetic field is described by the following equation:

The regions 1, 2, and 3 have vacuum permittivity *ε* = 1. The region “p” is a plasma layer of thickness *d* and permittivity $ \epsilon p = 1 \u2212 \omega p e 2 / \omega 2 $, where *ω _{pe}* is the electron plasma frequency. The last term in Eq. (1) describes the diffraction grating at

*x*= 0, where

*q*is the wave vector of the grating,

*μ*

_{1}is the modulation parameter, and

*μ*

_{0}is the mean (average) transparency of the grating.

^{24}This simple model was suggested in Ref. 24, and it is the simplest version of a more general model used in Ref. 25.

Equation (1) is used in the neighborhood of the diffraction grating to obtain the following boundary condition across the interface at *x* = 0:

The magnetic field is continuous at the grating, $ [ B z ] \u2212 \delta \delta = 0 $, where

We will use the notation in (3) to describe a discontinuity at any interface.

The diffraction grating leads to the generation of side-band harmonics with shifted wave vectors *k _{y}* ±

*q*. In general, there are multiple harmonics coupled via Eq. (2). Here, similarly to other works,

^{24,27,29,30}we neglect the higher order side-bands and consider the analytical solution in the form

Condition (2) generates the following relationships for the principal, *B*^{0}, and the sideband, *B*^{±}, harmonics:

where $ k g 0 = \omega 2 \epsilon g h g \mu 0 $/*c*^{2} and $ k g 1 = \omega 2 \epsilon g h g \mu 1 / c 2 $.

It follows from (1) that the derivative of the magnetic field is discontinuous at the plasma-vacuum interfaces. The relevant boundary condition is

The magnetic field is continuous at all interfaces, $ [ B z ] = 0. $

The boundary condition (7) is most conveniently written in terms of the impedances, Appendix A. The matching conditions (5,6) can be written in terms of the impedances as well by noting that

where the right-hand sides in these equations are evaluated at *x* = 0.

In this paper, we will consider only the case of normal incidence when *k _{y}* = 0 in Eq. (4). Then, the $ ( \xb1 ) $ sidebands become symmetric, $ B + = B \u2212 \u2009 \u2261 B \xb1 $, and one of the matching conditions at the diffraction grating can be written as

Using the definition of the impedance and the corresponding techniques described in Appendixes A and B, we find

The matching condition at the diffraction grating then becomes

This equation shows the coupling of the principal component and the side-bands at the diffraction grating. As in Ref. 24, to simplify the presentation, we set *h _{g}* =

*k*= 0 in Eq. (13). Note that a finite

_{g}*h*does not affect the surface wave resonances but introduces additional resonances of Fabry-Perot type which were considered in Ref. 3.

_{g}### A. The structure of the electromagnetic field and transmission coefficient

In this section, we describe the structure of the electromagnetic field in each region, formulate the matching conditions, and outline the calculation of the transmission coefficient. We will use the lower indices 1, 2, p, and 3 to define the relevant quantities in the corresponding region; we will use the upper indices (0, ±) to define the principal component and the sidebands.

#### 1. Vacuum region 1

In the outmost left vacuum region, we have the incident and reflected waves in the main harmonic

where *k*_{0} is the wave vector of the wave propagating in vacuum, $ k 0 2 = \omega 2 / c 2 $. Here, the amplitude of the incident was normalized to unity. Then, the running value of the impedance in this region is

The amplitude of the reflected wave $ \Gamma 1 0 $ can be related to the value of the impedance for the main harmonic in region 1 at the diffraction grating $ Z 1 0 ( 0 ) \u2261 Z 1 0 ( 0 \u2212 \delta ) , \delta \u2192 0 $

The characteristic impedance of vacuum for the principal harmonic is $ \kappa v 0 \u2261 k 0 c / \omega = 1 $.

The side-bands are evanescent in vacuum and are localized near the considered structure. Thus, in region 1, the side-bands are decaying as $ x \u2192 \u2212 \u221e $ and have the form

where $ \gamma v + 2 = q 2 \u2212 k 0 2 > 0 , \gamma v + > 0 $. The impedance of the sidebands in this region is constant

where *κ*^{+} is the characteristic impedance of vacuum for the evanescent side-band harmonics.

### B. Vacuum region 2

In the vacuum region 2, between the diffraction grating and plasma layer, the main and the side-bands harmonics have the following incident and the reflected components:

The corresponding expressions for the local values of the impedances are

Using (A5) and (A6), one can extend the impedances at the left and right boundaries in this region via the transformations

At the right boundary of region 2, across the vacuum-plasma interface, the impedances are continuous. Therefore, one can relate the impedances $ Z 2 0 ( a ) , \u2009 Z 2 + ( a ) $ in the vacuum region 2 to the impedances $ Z p + ( a ) , Z p 0 ( a ) $ in the plasma region as follows:

### C. Plasma layer region

In the plasma layer, principal components and both side-bands are evanescent

with $ ( \gamma p 0 ) 2 = \u2212 k 0 2 \epsilon p > 0 , \u2009 \u2009 ( \gamma p + ) 2 = q 2 \u2212 k 0 2 \epsilon p > 0 , \u2009 \epsilon p < 0 $. Note that that the coordinate system inside the plasma layer can be redefined so that the left side boundary of the plasma layer is at *x* = 0.

Then the local values of the impedances are

The impedances at the right and left boundaries of the plasma region are related by the transformations

where $ \kappa p \xb1 = i \gamma p \xb1 c / \omega \epsilon p , \u2009 \kappa p 0 = i \gamma p 0 c / \omega \epsilon p $ are the characteristic wave impedances in this region, and *d* is the width of the plasma layer.

At the right boundary of the plasma layer, the impedances are continuously matched to the vacuum region 3

We have dropped here the arguments for $ Z 3 0 $ and $ Z 3 + $ since the impedances in the region 3 are constant (see below).

### D. Vacuum region 3

In this region, we have only the transmitted wave propagating to the right and the side-bands that are decaying for $ x \u2192 + \u221e $

Respectively, the impedances in this region are constant and given by the relations

The transmission coefficient is given by the amplitude of the transmitted wave: *T *=* A*_{3} (see Appendix C).

## III. SURFACE WAVE RESONANCE AND REFLECTIONLESS TRANSMISSION

One of the most fascinating plasmonics phenomena is absolute transparency of an overdense plasma layer. Absolute transparency has been demonstrated in complex multi-layer structures where it can be supported by surface wave resonances as well as by standing wave (Fabry-Perot type) resonances.^{3} Some resonances caused by standing waves can be supported by additional modes which exist in warm plasmas.^{4} It was proposed in Refs. 2 and 24 that two diffraction gratings on both sides of a plasma layer are required to realize absolute transparency. This conclusion was based on the critical coupling concept.^{2,24} Here, we show that full transparency can also be achieved with a single grating.^{36} Using the impedance matching formulation, we show the role of the plasmon resonance and of the coupling to the standing wave resonance in region 1.

Full (100%) transmission is achieved when $ | T | = 1 $, where *T* is the transmission coefficient. This occurs when the reflected wave is absent in region 1: $ \Gamma 1 0 = 0 $ and therefore $ Z 1 0 ( 0 ) = 1 $. The impedances of the decaying sidebands in the vacuum region 1 are $ Z 1 + = Z 1 \u2212 = \u2212 \kappa v + $. Thus, the matching condition at the diffraction grating (13) has the form

Note that $ \kappa v + \u2009 $ is imaginary, and so is $ Z 2 + ( 0 ) $ for an ideal plasma; the impedance $ Z 2 0 ( 0 ) $ is in general complex. Then, for Eq. (39) to hold, the last term has to cancel out the imaginary part of $ Z 2 0 ( 0 ) $. For weak coupling, and small values of the parameter $ k g 1 2 c 2 / \omega 2 $, the resonance occurs near the pole of the last term in (39)

and the resonant condition is given by

where $ L p + = tanh ( \gamma p + d ) , \u2009 L v + \u2261 tanh ( \gamma v + a ) . $ It is easy to see that this is the exact dispersion equation for the surface wave of a plasma layer with a finite thickness *d*. The roots of this quadratic equation correspond to the symmetric and antisymmetric bonding of the surface waves localized on the opposite boundaries of the layer.

For a large thickness, we have $ L p + \u2192 1 $, and (42) yields

which is the dispersion equation for a surface mode at the interface of the vacuum and a semi-infinite layer with $ \epsilon p < 0. $ From (43), one finds the resonant value of the diffraction grating wave vector

In addition to the surface wave resonance described by Eq. (43), a second condition has to be satisfied for Eq. (39) to hold, namely,

This condition reduces to the following equation:

or equivalently

with $ L v \u2009 \u2261 \u2009 tan \u2009 ( k a ) , \u2009 L p = tanh ( \gamma p d ) $, and $ \kappa p = \u2212 i \beta $, where $ \beta = 1 / \u2212 \epsilon p $ is a real number. For given values of plasma permittivity $ \epsilon p $ and *L _{p}*, Eq. (47) has two roots

*L*which define the resonance condition for the distance between the plasma layer and the diffraction grating. For a thick plasma layer, we have $ L p \u2192 1 , \u2009 L v \u2243 1 / \beta $. In this limit, the resonance condition is given by

_{v}This condition indicates that the resonances are close to the standing wave resonances in the vacuum region 2: *ka *=* nπ*, where *n* = 1,2,…. In the general case

and the resonance value of the width of the vacuum region 2 is given by

Note that condition (47) does not depend on the diffraction grating parameters.

For finite values of the parameter $ k g 1 2 c 2 / \omega 2 $, the exact conditions for resonant transmission are obtained from (39) in the form

Conditions (45) and (51) have to be satisfied simultaneously for absolute (100%) transmission with $ | T | = 1 $.

As a numerical example, we consider the case of plasma parameters which are of interest for the communication blackout problem:^{37,38} $ \omega = 2 \pi \xd7 10 9 $ rad/s, plasma density $ n = 4.5 \xd7 10 17 $ m^{−3}, and the plasma layer width *d* = 0.02 m. In our calculations, we set $ \omega p = 2 \pi \xd7 6 \xd7 10 9 $ rad/s, which yields $ \epsilon p = \u2212 35 $. For these parameters, one finds from (49) and (50) the resonance value for the distance between the diffraction grating and the plasma layer *a* = 0.068153 m. According to Eq. (44), the resonance value of the diffraction grating wave vector *q* when the surface wave resonance occurs is $ q 0 = 21.2 \u2009 71 $ m^{−1}. The solution of the exact Eq. (51) gives two values for *q*, as seen in Fig. 2. Those values are slightly different from *q*_{0} due to coupling of plasmon modes in the plasma layer of finite thickness *d* [see Eq. (42)]. Figure 2 shows the reflection and transmission coefficients, each as a function of the diffraction grating wave vector for $ k g 1 2 = 4 \xd7 10 3 \u2009 m \u2212 2 $. The decrease in the parameter $ k g 1 $ results in the narrower resonance curves as shown in Fig. 3 for $ k g 1 2 = 4 \xd7 10 2 \u2009 m \u2212 2 . $

## IV. REFLECTIONLESS ABSORPTION IN A DISSIPATIVE PLASMA LAYER

The plasmon resonances that lead to the amplification of evanescent modes and absolute transmission through an overdense plasma layer are quite narrow in frequency (and the diffraction grating wave vector values). As such they are sensitive to dissipation and can be easily destroyed by dissipation in plasmas. At the same time, the same plasmon resonance is responsible for another interesting phenomenon that occurs due to interaction of an electromagnetic wave with an overdense plasma layer with dissipation, namely, the reflectionless absorption of the electromagnetic wave.

When electron collisions are included, the dielectric plasma permittivity becomes complex

The impedance matching formalism presented in Sec. II still remains valid and the same equations can be used to find the reflection and transmission coefficients as summarized in Appendixes B and C. In the dissipative case, $ | \Gamma 1 0 | 2 + | T | 2 < 1 $ since a fraction of the energy is absorbed inside the plasma. One can introduce the parameter $ A = 1 \u2212 | \Gamma 1 0 | 2 \u2212 | T | 2 $ to characterize the absorption. Normally, for overdense plasma with relatively weak dissipation, the absorption is low and most of the energy of the incident wave will be reflected, $ | \Gamma 1 0 | 2 \u2243 1 $. Near the plasmon resonance, however, the reflection can be reduced significantly.^{8,13,25,32} Full absorption was shown experimentally in Ref. 13.

The conditions for full absorption can be investigated analytically from results of Sec. II and Appendixes B and C as follows. Consider a sufficiently thick plasma layer so that transmission can be neglected, *T* = 0. In this limit, the entrance impedances of the vacuum region 2 become

For *T* = 0, the condition for reflectionless absorption is given by Eq. (39). The dominant part of $ Z 2 0 ( 0 ) $ is imaginary, and this part is compensated by the imaginary part of the last term in (39). It is worthwhile to note that in the dissipative case, $ \kappa p 0 $ acquires a small real part (related to the dissipation in plasma), $ \kappa p 0 = \u2212 i \beta + \alpha $, where *α* and *β* are real, and $ \alpha \u226a \beta $. The small perturbation of $ Z 2 0 ( 0 ) $ due to *α* is crucial as it generates the real part of $ Z 2 0 ( 0 ) $ which is required for the absence of reflection: $ R e [ Z 2 0 ( 0 ) ] \u2243 1 $; otherwise, $ Z 2 0 ( 0 ) $ is imaginary and the reflection is large. The condition $ R e [ Z 2 0 ( 0 ) ] \u2243 1 $ determines the resonant value of the distance from the diffraction grating to the plasma

Note that the condition $ R e [ Z 2 0 ( 0 ) ] \u2243 1 $ is further modified by the contribution from the last term in (39).

The exact condition for reflectionless absorption that follows from (39) has the form

where $ K = k g 1 2 c 2 / 2 \omega 2 . $ The $ ( \kappa v + + \kappa p ) $ term in the denominator of this expression emphasizes the role of the plasmon resonance [compare with Eq. (43)]. Equation (56) fully determines the exact conditions for the values of *a* (width of the vacuum region) and *q* (diffraction grating wave vector) for reflectionless absorption, when the energy of the incident electromagnetic wave is fully absorbed in the plasma, $ T = 0 , \Gamma 1 0 = 0. $ Plasma dissipation results in a slight modification of the resonant value of the diffraction wave vector compared to the values from (44) (see Fig. 3). An approximate value of *q* is found from Eq. (55). The dependence of the reflection and absorption coefficients as functions of the wave vector *q* is illustrated in Fig. 4. The decrease in the parameter *k _{g}*

_{1}results in the narrower region of absorption, as can be seen by comparing Figs. 4(a) and 4(b).

## V. SUMMARY

The phenomenon of anomalously large transmission of the electromagnetic waves through the media with negative dielectric permittivity such as dense plasmas (or metal films in the visible range) has attracted great interest in the last decade. It is generally understood that such transmission occurs as a result of resonant excitation of plasmon type surface modes. Resonant conditions can be created either with multi-layer zero- $ \epsilon e f f $ structures or with subwavelength structures. These subwavelength structures can be created by either a single defect such as a subwavelength aperture or with periodic arrays of holes or slits (diffraction grating). Resonant excitation of plasmon modes is a crucial element of many phenomena, such as reflectionless absorption, total backward reflection,^{39} negative refraction,^{1,2} and other phenomena in plasma based metamaterials.^{15,40}

In this paper, we have investigated resonant conditions and coupling of resonances in a system with an overdense plasma layer and a single diffraction grating. We have shown that extraordinary transmission also exists in this configuration, and not only in the two gratings scheme (on both sides of the plasma layer) proposed in Ref. 24. We have presented a simple impedance formulation that allows us to demonstrate clearly the coupling of the plasmon (evanescent) mode inside the plasma layer and standing wave resonances in the vacuum regions. We have demonstrated that those resonances lead to extraordinary transmission and reflectionless absorption. We have determined analytically the resonance value of the width of the vacuum region for full transmission [see Eq. (49)] and for reflectionless absorption [see Eq. (55)]. The exact expressions for the resonant values of the diffraction grating wave vector for full transmission are given by (51) and for reflectionless absorption by (56).

The impedance model presented in this paper is an exact alternative to the phenomenological critical coupling model proposed by others.^{13} The impedance model can easily be generalized to configurations with multiple layers and is convenient for numerical calculation of the reflection and transmission coefficients for continuous inhomogeneous plasma profiles.^{41–43} As such it is envisaged that the impedance model may become an effective tool for the investigation and design of plasma based metamaterial structures and configurations,^{40,44} including multi-layer and 2D structures for broadband applications.^{31} Plasma production and heating may be improved in plasma discharges designed with the explicit account of plasmon resonances.

## ACKNOWLEDGMENTS

This work was supported in part by AFOSR #FA9550–07-1–0415 and #FA9550–15-1–0226.

### APPENDIX A: IMPEDANCE

To clarify the notation, we now give the definition of the impedance and the formulas for the impedance transformations in a layer of finite length with permittivity *ε*. The impedance, *Z*, is defined by

In general, there is an incident wave $ \u223c exp ( \u2212 i \omega t + i k x ) $ and a reflected wave $ \u223c exp ( \u2212 i \omega t \u2212 i k x ) $, so that the magnetic field of the TM wave has the form

where *ω* > 0 and *k* > 0 are set so that the incident wave propagates to the right, and the reflected wave propagates to the left, with Γ being the reflection coefficient.

Thus, the local impedance is a function of the position *x*, and is given by

where *κ* is the characteristic wave impedance in the layer $ \kappa \u2261 k c / ( \omega \epsilon ) $.

The reflection coefficient is determined by the mismatch between the characteristic impedance *κ* and the value of the entrance impedance *Z*(0)

Equations (A3) and (A4) yield the standard transformation of the impedance for a finite interval of length *l*

The inverse transformation is then

These expressions can also be used for evanescent waves by setting *k *=* iγ*, with real-valued $ \gamma \u2260 0 $, so that

Then the characteristic impedance $ \kappa = i \gamma c / ( \omega \epsilon ) $ is imaginary, and the transformation relations (A5) and (A6) become

### APPENDIX B: SUMMARY OF THE IMPEDANCE MATCHING EXPRESSIONS

Here we summarize the nomenclature of the impedance matching relations and definitions that allow for sequential calculation of the reflection coefficient which can also be generalized for multi-layer structures. In our notation, the subscripts “1,” “2,” and “3” refer to the vacuum regions, and the subscript “*p*” refers to the plasma layer. The superscript “0” refers the principal harmonic, and the superscript “+” refers to the side-band, and the parameters $ \kappa v 0 $ ( $ \kappa p 0 $) and $ \kappa v + $ ( $ \kappa p + ) $ are the characteristic impedances for the vacuum (plasma) regions for the principal and side-band harmonics.

The impedance matching starts from the outmost right region 3 (vacuum) and proceeds to the left

The reflection coefficient is finally calculated from $ Z 1 \u2009 0 ( 0 ) $

For multi-layer structures, this process can be continued to the left. This method also offers a convenient way to calculate the total reflection coefficient in the case of a continuous profile by breaking it down into a number of sub-layers and repeating the cycle of calculations from the right to the left.

### APPENDIX C: CALCULATION OF THE TRANSMISSION AND ABSORPTION COEFFICIENTS

The sequence of calculations can be performed from left to right to obtain the transmission coefficient. This sequence is summarized here

Here, according to (A4)

The transmission coefficient is *T *=* A*_{3}, and therefore is given by

For ideal plasmas, $ | \Gamma 1 0 | 2 + | T | 2 = 1 $, and it is sufficient to calculate only one coefficient. In plasmas with dissipation, both coefficients $ \Gamma 1 0 $ and *T* have to be determined independently. The energy absorption can be characterized by the parameter *A*, defined as $ A = 1 \u2212 | \Gamma 1 0 | 2 \u2212 | T | 2 $.