Resonant absorption is a process that electromagnetic (EM) energy is converted to plasma energy with a mode conversion on the resonant layer where the incident EM wave frequency equals to the local frequency of a plasma normal modes. With a finite collision between charged and background neutral particles in a plasma, the plasma oscillation is dissipated to widen the resonance layer and heat the plasma. In this work a modified scattering matrix methods (SMM) are applied to study the effects of the collision frequency, incident angle, and plasma thickness on resonant absorption. We analyze the energy absorption caused by resonance in comparison with collisional absorption for different parameters. It is found that the resonant absorption dissipates about nearly half of the incident EM energy in an overdense inhomogeneous plasma when the collision is weak, and the rest half portion is reflected. If the collision is strong, however, the collisional absorption is then more significant than the resonant and affects the entire wave propagation process.

Propagation properties of electromagnetic (EM) waves in sub-wavelength plasma slabs are getting more attentions in recent years due to its simplicity for understanding features of wide applications in microwave diagnostics,1 plasma-wave interactions,2,3 and so on. With an inhomogeneous density profile, if the plasma slab thickness is comparable to or less than the incident EM wavelength, resonant absorption may occur and cause transmission and absorption of waves.4,5

The occurrence of resonant absorption of EM waves in inhomogeneous plasmas can be found where the frequency of incident wave equals to the electron plasma frequency.6,7 In the extremely thin resonant layer, the electric field has an approximate singularity if the collision can be ignored. This property is often applied to metamaterials as wave concentrators,8,9 polarization manipulation,10 and so on. If the collision between electrons and background neutral particles is nonnegligible, the incident EM energy will be transformed to kinetic energy of electrons and be absorbed by plasmas,11,12 thus the plasma will be heated.13 Considering the significant absorption caused by resonance, inhomogeneous plasma layers or metamaterials may also be used to cloaking.14,15 Thus, resonant absorption has extensive applications not only in plasma physics but also in the specific materials designing.

Resonant absorption was studied usually under the cold plasma assumption with temperatures of electrons and ions are neglected, and also in the geometric-optics approximation where the plasma size is macroscopic with the density varying slowly.16,17 It was found that the electric field in the plasma was observably amplified in the singular layer where the plasma frequency equals to the frequency of the incident EM wave. Resonant absorption in hot plasmas was also studied, particularly with nonlinear density modifications by intensive EM waves or lasers.18–20 In the singular resonance layer, a mode conversion process21–23 was put forward to describe how the incident EM waves transform to plasma normal modes. Local net charge and electrostatic (ES) field singularities were found in the resonant layer. Similar resonant absorption processes also occurred in grad-refraction-index materials.24,25 Increasing the index of metamaterials gradually from positive to negative, the mode conversion appeared in a thin layer where the permittivity or permeability almost vanished.26–28 It was found that the P-polarized electric field of the incident wave increased if the permittivity of the material was close to zero, while the S-polarized magnetic field of the incident wave enhanced if the permeability is close to zero. Thus, if profiles of the permittivity and/or permeability were carefully designed, both electric and magnetic fields of the S- and/or P-polarized might be amplified in the singular layer. Influences of different permittivity or permeability profiles on the propagation of EM waves with resonant absorption were also studied.20 It was found that plasma with a linear density profile has higher total absorption for incident EM wave than that with parabolic and exponential profiles.

In previous works mentioned above, fundamental mechanisms and typical applications of resonant absorption were summarized. There nevertheless are still certain important aspects left. Particularly, effects of the collision on the mode conversion and energy absorption in the resonant absorption layer should be investigated further in certain applications. Besides, the inhomogeneity and sub-wavelength thickness of the plasma slab make it hard to calculate using traditional algorithms such ray tracing.29 The terminology of “sub-wavelength” in the literature in general means the scale of the medium is less than the wavelength of the incident EM waves. In our case then, the medium size is the plasma thickness. In this paper, we first briefly analyze the mode conversion from the EM wave to the plasma normal modes at the resonance. Then modified scattering matrix methods (SMM)30,31 are applied to simulate the process, and analyze effects of the collision, incident angle, and plasma slab thickness on waves in subwavelength plasma slab. Furthermore, the ratio of the resonant absorption to the incident EM energy is also calculated.

Consider a one-dimensional plasma slab as shown in Fig. 1, where the plasma density is non-uniform in the x-direction and uniform in the y- and z-directions. The thickness of the slab is D, and the slab extends infinitely in the transparent y- and z-directions. The EM wave is polarized in the incident plane (x, y), with a frequency f0 and an incident angle θ.

FIG. 1.

Schematic of the plasma slab and the incident EM wave.

FIG. 1.

Schematic of the plasma slab and the incident EM wave.

Close modal

In a typical ICP discharge plasma or plasmasonic sheath, the plasma density is around 1017m-3, then the ion response time is much longer than the period of the incident microwave with frequency around 1GHz, only the electron response should be taken into account. Thus we have the continuity and the moment equation for electrons

net+neue=0,
(1)
nemeuet+ueue=peneeE+ue×Bcνennemeue,
(2)

as well as the state equation of ideal gas pe=neTe. The current density is defined as

J=nieuineeueneeue.
(3)

Here, Gaussian unit system is applied, ni and ne are the ion and electron number densities, ui and ue are the ion and electron fluid velocities, pe is the electron thermal pressure, Te is the electron temperature, me is the electron mass, e is element charge, c is light speed and νen is the collision rate between electrons and neutral particles. In this paper, we mainly focus on the weakly ionized plasma generated under sub-atmospheric or even atmospheric pressures. With such relatively high pressures, the collision of the electron-neutral is much more frequent than that of the electron-ion or between electrons. Sometimes, the electron-neutral collision frequency can be as high as the incident EM wave frequency,32 and thus has significant effect on the electron momentum and cannot be neglected in equation (2). While collision frequencies of the electron-ion and between electrons are much lower than that of incident waves, they then have little impact on election motions.

Maxwell’s equations are also utilized as follows

×B=4πcJ+1cEt,
(4)
×E=1cBt,
(5)
E=4πenine.
(6)

In linear approximations, we have ue = ue1, E = E1, ne = ne0 + ne1, and ni = ni0 = ne0. For a P-polarized wave, we have Bx = By = 0 and Ez = 0. Assuming that Bz, Ex and Ey vary by time with an oscillation factor of eiω0t, one can obtain from equations (1)–(6) that4 (The details are shown in  Appendix.)

2Bzx2nxn1+iνnBzx+1n1+iνα2Bz   =nxα2β21+iνnExx,
(7)
2Exx2nxExx+1+iνnβ2Ex   =αnx1+iν1+iνnBzx1+iνβ2Bz,
(8)

where α = sin θ, β=Te/mec2, n = (ωpe/ω0)2 = ne0/nc, nc=meω024πe2, ν = νen/ω0 and x is rescaled by xk0x = ω0x/c, with ωpe=4πne0e2/me is the plasma frequency and ω0 is the angular frequency of the incident wave. When the EM wave is perpendicularly incident to the plasma slab, i.e., α = 0, Bz and Ex are decoupled. Further, by assuming the plasma is uniform, Eq. (7) becomes a typical wave equation for the EM wave in plasmas with a dispersion relation of

n=1+iν1kx2,
(9)

where ikx=/∂x. Equation (9) represents an ordinary wave if the collision is ignored. From Eq. (8) on the other hand, we obtain the dispersion relation for Ex as

n=1+iνkx2β2,
(10)

which is apparently a Langmuir wave (i.e., the plasma wave hereafter) propagating in the x-direction. We can see that the ordinary EM and Langmuir waves are independent of each other when the incident angle is zero. Nevertheless, in fact, except on an interface of different media, both kx and Ex are vanished when the incident angle is zero. Thus, the plasma wave cannot be observed in the plasma. With a nonzero incident angle however, kx and Ex become finite and the plasma wave is excited. Nonetheless, the plasma wave will not be excited anywhere in the plasma, but a special layer. Comparing relations (9) and (10) we can find that they intersect one another at n = 1 (ω0 = ωpe) if the collision is ignored. At this point the incident EM wave, represented by (Ey, Bz), will be partially transformed to an induced plasma wave represented by the ES Ex propagating in the x-direction with a phase velocity βc. However, if the collision is significant or α ≠ 0, relations (9) and (10) is inadequate to describe modes in plasmas and solving Eqs (7) and (8) is necessary.5,22,23

In low temperature plasmas, the condition β=Te/mec2υe/c1 is always satisfied. Thus, the plasma wave has a negligible phase velocity and the electrons only oscillate locally in a frequency of ω0 (= the local ωpe) with no propagation while the EM wave is not affected. Eqs. (7) and (8) can then be simplified to

2Bzx2nxn1+iνnBzx+1n1+iνα2Bz=0,
(11)
2Exx2nx11+iνnExx+1n1+iνα211+iνn2nx2   11+iνnnx2Ex=0.
(12)

Here Ex includes both EM and ES contributions. Our simulation using the modified SMM is then based on these Eqs. (11) and (12).

If the collision is ignored, the coefficients in equation (12) become infinite at the singular layer where n = 1, and the electric field also gets infinite, though in the simulation however, both of them are large but finite due to the numerical dissipation. We also note that the singularity disappears if the plasma is uniform. Then the nonuniformity of plasma is a necessary condition to excite the resonant mode conversion.

If the collision is considered, the singularity of Eq. (12) becomes resolvable with the electric field at the singular point becomes finite. In fact, Eq. (12) is also resolvable when the electron temperature effect is taken into account.5 From Eq. (4) we find that the maximum electric field can be written as

Exmax=αBz01+iνn1iν=1+ν2νBz0 sin θ,
(13)

where Bz0 is amplitude of the magnetic field at the singular point which is always less than that of the incident magnetic field. We may find that maximum electric field |Ex|max is comparable to |Bz0| when the collisional rate is approximately the frequency of the incident wave, i.e., ν > 0.5. In other words, for the strong collision plasmas, the maximum electric field is mainly EM, and the ES field due to the mode conversion is greatly reduced. Near the singular point x = x0, the electron density profile can be linearized, i.e., as ne0nc = (x - x0) dne0/dx (where nc = ne0(x0)), the full width at half maximum (FWHM) of the electrical field peak can be estimated by Eq. (13) as

Δν/2,
(14)

normalized by the wavelength of the incident wave. We then find that the resonant layer is broadened by the collision.

In the SMM,30,31 the plasma slab is divided into a serial of sublayers, and by slightly modifying the method, Bz in each sublayer can be written as

Bzm(x)=(ameiNmx+bmeiNmx)eikyy,

where am and bm are amplitudes of incident and reflected fields in the m-th sublayer, and ky is the wave vector in the y-direction, and Nm is the refractive index in the m-th sublayer defined by

Nm=1nm1+iνα2,
(15)

where nm is the normalized plasma density (or the square of the normalized plasma frequency) in the m-th sublayer. Then the outgoing magnetic field is

aoutbout=SM1SM2S1a1b1=Sa1b1,

where

Sm=11Nm+11+iνnm+1Nm+11+iνnm+1111Nm1+iνnmNm1+iνnmeiNmΔxeiNmΔx.

Here Δx is the size of each sublayer. From Eq. (4) we get the peak electric field

Exm=sin θ1+iνn11+iνnmameiNmx+bmeiNmxa1eikyy,
Eym=cos θNmN11+iνn11+iνnmameiNmxbmeiNmxa1eikyy.

Then the reflection coefficient r and transmission coefficient t are

r=b1a1=S21S22
t=aMa1=S11S22S12S21S22

We define the reflectivity R and transmissivity T as follows

R=r2,
T=t2.

Here Sij is the element of Matrix S. And the absorptivity is A=1-R-T. Inspecting the algorithm above, the only assumption in SMM is that the plasma density is uniform in each sublayer which has a tiny thickness, thus the main errors are caused by discretization. Comparing with ray tracing method, modified SMM avoids WKB approximation, hence can be employed on plasma slab with subwavelength thickness, where the plasma density varies sharply by location. In fact, the thickness of a typical plasmasonic sheath is around 10cm32 which is less than wavelength of telemetry signal with frequency around 1GHz. Thus an algorithm based on full wave theory, such as SMM, will be necessary in such cases.

In our simulations, the plasma density profiles are assumed a combination of two half-parabolas as

n=nmax1(x/d11)20x<d1nmax1(xd1)/d22d1xD,

where D = d1+d2 is normalized by λ0 = 2π/k0, nmax is the maximum electron number density in the plasma slab. The shape parameter is defined by η = d1/d2. Various density profiles with different shape parameters are shown in Fig. 2.

FIG. 2.

Profiles of the normalized density for different shape parameters η with D = 1, where η=0.1 in (a), η=1 in (b) and η=10 in (c).

FIG. 2.

Profiles of the normalized density for different shape parameters η with D = 1, where η=0.1 in (a), η=1 in (b) and η=10 in (c).

Close modal

The EM wave is incident from the left to the right side of the plasma slab. Using the solver above, we can study the propagation characteristics of the EM wave in the plasma slab with different parameters, including the plasma density, the collision rate, incident angle, the shape parameters, and so on.

To check the validity of modified SMM, we compare the results with FDTD results where the relative permittivity of plasma is

εr=1ωpe2ωωiνen=1n1iν.

Assume that the thickness of the plasma slab is D=1, the collision rate is ν=1 and the incident angle is 45°. The profile of plasma density is symmetric i.e. η=1. The results are shown in Fig. 3. The solid line is the transmission coefficient obtained from modified SMM and the upper triangles are the same quantity calculated by FDTD. It can be seen that the solid line agrees well with the upper triangles. Besides, according to our calculations, FDTD consumes more than tenfold time to achieve the same accuracy comparing with modified SMM. Thus we conclude that modified SMM is a more applicable means to simulate waves in plasma slabs than FDTD.

FIG. 3.

Comparison of the transmission coefficients between modified SMM and FDTD, where D=1, ν=0.01, θ=45, η=1.

FIG. 3.

Comparison of the transmission coefficients between modified SMM and FDTD, where D=1, ν=0.01, θ=45, η=1.

Close modal

Case I: A thick plasma slab of D = 20.

For a thick plasma slab with D = 20 in the geometric-optics regime, profiles of |Bz|, |Ex| and |Ey| are shown in (a), (b) and (c) of Fig. 4 respectively. The vertical dashed line marks the resonant point where n=1.

FIG. 4.

Profiles of |Bz| in (a), |Ex| in (b), and |Ey| in (c) in the plasma slab with D=20, ν = 0.01, nmax=2 and η = 10, with the incident angle θ = 10°.

FIG. 4.

Profiles of |Bz| in (a), |Ex| in (b), and |Ey| in (c) in the plasma slab with D=20, ν = 0.01, nmax=2 and η = 10, with the incident angle θ = 10°.

Close modal

From Fig. 4(a) we can find that the magnetic field rapidly decreases after the cut-off where n = 1 - α2. Before that point the magnetic field amplitude oscillates as a standing wave caused by interference between the incident and the reflected waves. From Fig. 4(b) a sharp peak of |Ex| is excited at the resonant point where n = 1. While there is no obvious peak of |Ey| in Fig. 4(c), indicating that the resonance is mainly caused by the ES field Ex perpendicular to the slab. That is the reason why we cannot see a resonance peak when the EM wave is perpendicularly incident to plasma,31 or the electric field of the incident wave is S-polarized with the electric field polarized in the z-direction.33 Furthermore, by Eq. (14) we find that the magnetic field Bz at the resonant point should not vanish at the point in order to excite the resonant mode conversion. Therefore, if the plasma density varies slowly or the incident angle is large, the separation between the cut-off and the singular points, i.e., s = α2dx/dn, will be too big to excite the resonant mode conversion, since the incident EM field will be significantly reduced after the cut-off point and can hardly reach the singular point.

Case II: A sub-wavelength plasma slab of D = 1.

It is obvious that resonant mode conversion is excited more easily when the plasma thickness is small. Fig. 5 shows distributions of the electric field amplitude for different plasma densities and collision frequencies with a sub-wavelength slab of D=1 (=2π/k0 = λ0). In the figure, the electric field includes the incident and the reflected fields, i.e. |E|=|am+bm|. Under the weak collision condition (ν = 0.1), we can see in Fig. 5(a) that the electric field amplitude decreases from 2 to 0 when incident angle is 0°, i.e., the incident EM wave is totally reflected. Increasing the incident angle, peaks of the electrical field then appear at n = 1, where the plasma frequency equals to the incident wave frequency. Obviously, those peaks are caused by resonance between the incident EM wave and the induced Langmuir oscillation. After leaping over the peak, the electric field amplitude is getting lower than that without resonance. For example, the electric field for θ=30° at x = 0.25 is smaller than that for θ=0° at the same position. It means that the electric field is strongly localized around the resonance point. Besides, the peak amplitude of |E| will increase to 3.3 with the increase of incident angle. The x component of the electric field, Ex, the main cause of resonance, is increased as the incident angle arises. Hence a strong resonance is obtained. If the incident angle continues to increase, the separation between cut-off and resonance (i.e., s = α2dx/dn) points, will be enlarged, and then the incident field will be reflected at an earlier cut-off point where n=1-α2 and can hardly reach the resonance. Thus, it is shown that the resonant peak of Ex becomes shorter. In Fig. 5(b) we can see that the resonance peak is also significantly affected by the electron-neutral collisional rate. When the rate ν = 0.01, the peak reaches 28 with an FWHM of 0.007, and then decreases to 3 when ν = 0.1 with the FWHM of 0.058. The resonance peak is apparently broadened by collisions with Δ ∼ ν as indicated in (15). Particularly, the resonance peak is smoothed down and disappears when ν =1. Continuing to increase collision frequency, the resonance peak never shows up. The reason is that the strong collision dissipating the electric energy around the resonance and reducing the peak while broadening the region. Analytically, with no collision, the electric field goes to infinite. In the simulation however, the peak reaches 104, indicating a very low numerical dissipation of ν ∼10-4 and a high resolution of Δ ∼ ν/2 for the solver.

FIG. 5.

Field distributions with nmax = 2, D = 1, η = 1, in (a) different incident angles when ν = 0.1; (b) different collision rates when θ=45°.

FIG. 5.

Field distributions with nmax = 2, D = 1, η = 1, in (a) different incident angles when ν = 0.1; (b) different collision rates when θ=45°.

Close modal

Then the peak values and the FWHMs of the resonant peaks of Ex are estimated under different collision frequencies and incident angles as shown in Fig. 5. It can be seen in Fig. 6(a) that |Ex|max is proportional to 1/ν when 1/ν > 2 (i.e., ν < 0.5). Then in this case Eq. (14) can be written as ExmaxBz0 sin θ/ν. Besides, |Ex|max varies with the incident angle. Obviously, the simulation results are in good accordance with Eq. (14). Fig. 6(b) shows the FWHM vs. collision rate ν under different incident angles. The red dashed line Δ = ν/2 is a very good fitting for those curves with ν < 0.25, as predicted in Eq. (15) for the weak or moderately weak collision regimes. While ν > 0.25, the FWHMs deviate from the fitted line due to the stronger collision. Therefore, we can deduce a criterion that the resonant absorption is considerable for ν < 0.25 under the condition that D=1, nmax=2 and η=1.

FIG. 6.

The peak value |Ex|max in (a) and the FWHM in (b) under different collision rates and incident angles where D=1, nmax=2 and η=1.

FIG. 6.

The peak value |Ex|max in (a) and the FWHM in (b) under different collision rates and incident angles where D=1, nmax=2 and η=1.

Close modal

The reflectivity, transmissivity, and absorptivity of incident EM waves for various collision rates and incident angles are shown in Fig. 7. (1) In the weak collision regime (ν < 0.1), the transmissivity always vanishes. By increasing the incident angle, the reflectivity will decrease first and then grows up again to unity, while the absorptivity is large when the incident angle is around 45° due to apparently intensive resonant absorption in that regime. (2) In the moderate to strong collision regimes (0.1 < ν < 10), the incident EM wave is almost totally absorbed. But if the incident angle is larger than 80°, i.e., the wave vector is almost parallel to the plasma surface, the incident wave is mostly reflected. (3) In the extremely strong collision regime (ν > 10), the EM wave is transmitted within a large parameter range.

FIG. 7.

(a) The reflectivity R, (b) the transmissivity T, and (c) the absorptivity A for D = 1, nmax= 2, η = 1.

FIG. 7.

(a) The reflectivity R, (b) the transmissivity T, and (c) the absorptivity A for D = 1, nmax= 2, η = 1.

Close modal

From equation (15) we have

N=1α2n1+ν2+inν1+ν2.
(16)

Firstly consider ν=0, then N=1α2n. On the incident side of cut-off point where n=1-α2, N is a real number and the wave propagates forward. While on the other side N will be imaginary and the incident field decays sharply thus be reflected. When collision rate is nonzero, cut-off point is where n=(1-α2)(1+ν2). If (1-α2) (1+ν2)>nmax, the incident wave will not be cut-off or reflected. Besides, because the imaginary term, i.e. inν/(1 + ν2), will be small when collision rate is extremely large, the incident wave will not be reflected or absorbed and plasma slab can be regarded as transparent in this case. Figure 8 shows the amplitudes of magnetic field in plasma slab when collision rate is 20 (ν=20). It can be seen that when incident angle is small (θ<80°), the amplitudes of magnetic field at the incident boundary are close to 1, which means the reflection is small in this regime where cut-off cannot be observed. While the incident angle is large (θ>80°), the incident wave is partially or mostly reflected. Note that the simulation criteria of cut-off above, i.e. θc=80°, is not exactly the same as the theoretical results because the plasma is inhomogeneous and has a subwavelength thickness in simulation. In fact, there is no clear criteria to determine cut-off process in simulation.

FIG. 8.

The amplitudes of magnetic field in plasma slab under extremely strong collision i.e. ν=20. Plasma thickness and density are D=1, nmax=2, plasma shape parameter is η=1.

FIG. 8.

The amplitudes of magnetic field in plasma slab under extremely strong collision i.e. ν=20. Plasma thickness and density are D=1, nmax=2, plasma shape parameter is η=1.

Close modal

The imaginary part of equation (16) is

NI=121α2n1+ν22+nν1+ν221α2n1+ν212,

which represents the absorption of incident wave by plasma. Apparently, NI has maximum value when 1 − α2n/(1 + ν2) = 0. Thus in theory, if θ<θc=arcsin1n/1+ν2, absorptivity will increase as the adding of incident angle. While θ>θc=arcsin1n/1+ν2, the absorptivity will deduce by incident angle. Figure 7(c) gives similar results when ν>10.

To further study the energy dissipation of the incident wave caused by resonant and collisional absorptions, we define the absorbed power per unit area in y-z plane in the m-th sublayer as

ΔWm=mΔx(m+1)ΔxσE2dx=nmν1+ν2mΔx(m+1)ΔxEm2dx,
(17)

where σ is the conductivity. The results are shown in Fig. 9. It can be seen that the EM wave power loss is mainly in the resonance layer when the collision is weak (ν < 0.1). While out of resonance layer where collision dissipation is the main reason to energy loss, power loss is small. In fact, it can be obtained from Fig. 9 that the ratio between loss power caused by resonance and total power loss is about 95.68% for ν = 0.01 which means nearly all the power loss of EM wave is in resonance layer when collision is weak. As the collision increases, the resonance layer is significantly broadened. In the case of ν ≥ 0.5, the resonance peak almost disappears and the maximum value of power loss moves away from the resonance point. Differing from resonant peak, those smooth peaks under strong collision is caused by interference between incident and reflected waves just like peaks in Fig. 4(a).

FIG. 9.

The distribution of the power loss in the plasma slab for different collision rates when nmax=2, D=1, η=1, θ=45°.

FIG. 9.

The distribution of the power loss in the plasma slab for different collision rates when nmax=2, D=1, η=1, θ=45°.

Close modal

Since nearly all the power loss is absorbed within a thin resonance layer under weak collision, we set fixed boundaries for resonant layer as shown by two dashed vertical lines in Fig. 9 to estimate the resonant absorption power, which means the energy loss in the boundaries is defined as resonant absorption power ΔWres and out of the boundaries the energy loss is regarded as collisional absorption power ΔWcoll. Here ΔWresWcoll) is summation of ΔWm in equation (17) for sublayers in (out of) resonant region. The reason why we choose the fixed boundaries is as follows. Consider weak collision, resonant peak will distribute in between the two boundaries. The defined resonant absorption power include both resonant and collisional absorption power which are hardly to be decoupled, but the collisional absorption in the thin resonant layer is negligible comparing with resonant absorption when collision is weak. Thus ΔWres can be used to represent resonant absorption. Besides, this estimation will be precise when collision rate is small, where the FWHM of resonant peak varies little by collision rate. However, if the collision is strong, the range of resonant peak will distribute out of defined boundaries, and collision absorption will be important. As to these cases, ΔWres will be not a proper estimation for resonant absorption. Here, we mainly concern about the energy absorption caused by resonant process when collision is weak in this paper. Therefore, the assumption and estimation of ΔWres is reasonable and acceptable under such conditions. Inspecting the profiles of power loss in Fig. 9 under different collision rate, we artificially set the separation between two boundaries be dres=0.03, then resonant peaks will totally distribute in the boundaries when ν<0.05. Then the absorptivity A can be divided into two parts, i.e., the resonant absorption rate Ares and collisional absorption rate Acoll, which are defined as

Ares=AΔWres/ΔWtot,
Acoll=A1ΔWres/ΔWtot.

The resonant absorption rates under different incident angles when collision is weak are shown in Fig. 10. Apparently both the collision rate and the incident angle have significant effects on resonant absorption. When the collision rate is small (ν < 0.05), the absorption of the incident wave is mainly caused by resonance. Besides, as the incident angle increases, the resonant absorptivity rises first and then drops to zero, with a maximum value of 42%. Most EM wave power is then reflected within this regime as shown in Fig. 7(a). As be stated above, x- or normal component of the electric field is necessary to excite resonant mode conversion. Since Ex will increase when the incident angle is added, then resonant absorption will rise too. Continuing to increase the incident angle, the EM field reaching the resonant point is reduced, thus the resonant absorptivity diminishes accordingly. Hence the incident angle should be in an appropriate regime to excite a strong resonance.

FIG. 10.

The resonant absorptivity Ares in (a) and the collisional absorptivity Acoll in (b) for different collision rates and incident angles with nmax=2, D=1, η=1.

FIG. 10.

The resonant absorptivity Ares in (a) and the collisional absorptivity Acoll in (b) for different collision rates and incident angles with nmax=2, D=1, η=1.

Close modal

Fig. 11 show the resonant, and the collisional absorption rates for different shape parameters. It can be seen that the resonant absorption rate will increase when the shape parameter is decreased. As is shown in Fig. 6(a), the resonant peak value is proportional to the magnetic field at the resonant point. When plasma density profile on the incident side becomes sharp, separation between cut-off and resonant point, which can be estimated by s = α2dx/dn, will deduce and then the magnetic field at the resonant point grows up, causing a stronger resonant absorption. To excite strong resonance process, the plasma density profile should be sharp enough.

FIG. 11.

The resonant, Ares in (a), and the collisional, Acoll in (b) absorption rate for different collision rates and shape parameters with nmax=2, D=1, θ=45°.

FIG. 11.

The resonant, Ares in (a), and the collisional, Acoll in (b) absorption rate for different collision rates and shape parameters with nmax=2, D=1, θ=45°.

Close modal

Fig. 12 shows the resonant, and the collisional absorption rates for different plasma slab widths. We can see that the resonant absorption is larger than 0.3 when 0.3 <D<2.5 under the condition of weak collision. When plasma slab width is thick, the slope of the plasma density profile around will be small, thus the EM field will be reduced sharply after passing the cut-off point and not reach the resonant point. While the plasma slab width is very thin, a second resonant peak also satisfying n=1 will appears. In reality, the incident wave will be mostly transmitted when the plasma slab is pretty thin, thus the resonant absorption is weak within this regime.

FIG. 12.

The resonant, Ares in (a), and the collisional, Acoll in (b) absorption rate for different collision rates and plasma thicknesses with nmax=2, η=1, θ=45°.

FIG. 12.

The resonant, Ares in (a), and the collisional, Acoll in (b) absorption rate for different collision rates and plasma thicknesses with nmax=2, η=1, θ=45°.

Close modal

Mode conversion with resonant absorption is a process that EM wave is transformed to ES plasma mode and then be dissipated by collisions between electrons and neutral particles. Around the resonant point, ES field is significantly amplified and the magnetic field is weak, thus the ES field is converted from the electromagnetic field.

The resonant absorption process is affected by the collision rate, incident angle, plasma slab width and plasma density profile. Under the condition that D=λ0, nmax=2nc and η=1, when the collision rate is small (νen < 0.05f0), resonant absorption plays a dominant role in the propagation of the EM wave. The maximum resonant absorption power is around 42% of the total incident power in an overdense inhomogeneous plasma. The rest part of the EM wave power is reflected. By increasing the collision rate, the resonant peak will be smoothed down and the resonant absorptivity will decrease. When collision rate is large (νen > 0.25 f0), resonant absorption nearly disappears and collisional absorption could be the main cause of the power loss and the maximum collisional absorptivity reaches unity. When collision rate is moderately weak (0.05 f0 < νen < 0.25 f0), both resonant and collisional absorptions are important. By increasing the incident angle from 0° to 90°, the resonant absorptivity will increase first and then diminish to zero, and most of the EM field is reflected by the overdense plasma slab even if the resonant absorption is very strong. Besides, increasing the plasma density slope on the incident side, resonant absorption rate will be aggrandized. Because the separation between cut-off and resonant point decreases and the incident can easily reach the resonant point after pass through the cut-off point. When thickness of the plasma slab is around the wavelength of the incident wave, the resonant absorptivity will be considerable. But if the plasma thickness is large under parabolic profile with the same peak value, the slope of plasma density will become slow and the incident field will be mostly reflected, resulting in a small resonant absorption.

This work has been supported by National Natural Science Foundation of China (No. 11875118, 41674165) and Science Challenge Project (No. JCKY2016212A505).

From equations (2)–(6) we obtain that

EyxikyEx=iωcBz,
(A1)
Exx+ikyEy=4πen1,
(A2)
ikyBz=4πn0ecuxiωcEx,
(A3)
Bzx=4πn0ecuy+iωcEy,
(A4)
n0meνiωux=Ten1x+n1Telnn0n0eEx,
(A5)
n0meνiωuy=ikyTen1n0eEy,
(A6)

where n0 and ne0 denote n1 and ne1 for simplicity, ky=k0sinθ. From equation (A2), (A5) and (A6), we have

ux=Te4πn0meeνiω2Exx2+ikyEyx ln n0Exx+ikyEyemeνiωEx
(A7)
uy=ikyTe4πn0meeνiωExx+ikyEyemeνiωEy
(A8)

Substitute (A7) and (A8) into (A3) and (A4) respectively, we have

kycωBz=β2c2ωω+iν2Exx2+ikyEyx ln n0Exx+ikyEy1ωpe2ωω+iνEx,
(A9)
Bzx=kyβ2cω+iνExx+ikyEy+iωc1ωpe2ωω+iνEy
(A10)

Equations (A1), (A9) and (A10) are equations of Ex, Ey and Bz. Get the expression of Ey from (A10) and substitute it into equation (A1) and (A9), Ey will be eliminated and then we have equations (7) and (8). Note that there is no assumption in above derivations.

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