St. Petersburg State Electrotechnical University
共Submitted January 29, 1999兲
Pis’ma Zh. Tekh. Fiz. 25, 50–55共December 12, 1999兲
An analysis is made of stable and unstable surface electromagnetic waves at the boundary of a planar moving plasma layer. It is shown that unlike an isolated tangential velocity
discontinuity, both slow and fast waves may exist inside the layer. The maximum spatial growth rate of the oscillations is achieved for directions of wave propagation other than the
direction of motion of the layer. Symmetric and antisymmetric waves relative to the symmetry plane of the layer have different critical angles from which their growth evolves and the
range of angles where the flux is stable is determined by the smaller of these. © 1999 American Institute of Physics.关S1063-7850共99兲00812-5兴
The formation of a surface electromagnetic wave near the planar interface between two media is related to the ex- istence of an imaginary part of the surface impedance1which corresponds to the case where one of the media has a com- plex refractive index.2In the absence of losses, this situation may arise if the permittivity is⬍0, i.e., for a plasma. When both fixed boundary media have positive permittivities, it is impossible for surface waves to be excited near the interface. The relativistic motion of one of these media leads to the possible excitation of an electromagnetic surface wave even at a tangential velocity discontinuity in a homogeneous me- dium with ⬎0. In this case, a necessary condition for the existence of surface waves is the formation of anisotropy as a result of the appearance of a selected direction of motion of the medium.4Then, the waves can only propagate in direc- tions forming angles greater than some critical angle with this direction of motion.
In addition to stable waves, growing surface electromag- netic waves also exist at a tangential velocity discontinuity, which leads to hydrodynamic instability of the relativistic plasma fluxes.5For a planar layer, relativistic motion relative to a stationary external medium promotes instability as a result of waves moving away from the interface being ex- cited in the external dielectric medium 共Cˇerenkov instability兲6 or surface waves if the stationary medium is a
plasma.7 However, the analysis reported in Refs. 6 and 7 only referred to the case of a collinear velocity and wave vector.
We shall consider a moving planar plasma layer of thickness 2h having the permittivity 2 and velocity
V⫽c in a stationary medium having the permittivity1. In rectangular coordinates the velocity has the form ⫽(0,y,z), and the wave vector k⬜is directed along the z axis. From Maxwell’s equations for a moving medium and boundary conditions we obtain:
共21S⫹12兲共1S⫺1⫹2兲 ⫽共2⫺1兲共1⫺1兲2␥2y 2 , 共1兲 where 1 2⫽2⫺ 1, 2 2⫽2⫺1⫺( 2⫺1)␥2(1⫺z)2, ⫽kz/k, k⫽/c, kz⫽兩k⬜兩, and␥⫽(1⫺2)⫺1/2.
Equation共1兲 is the dispersion equation for surface waves in a planar waveguide filled with a moving plasma, and de- scribes waves having symmetric 关S⫽coth(kh2) 关and anti- symmetric 关S⫽tanh(kh2)兴 components of the field Ez rela- tive to the y z plane. For a tangential velocity discontinuity in a homogeneous medium i.e., 1⫽2, the equations 共1兲 for the symmetric and antisymmetric waves are the same. For
FIG. 1. Propagation constant kzas a function of the angle
between k⬜ and V for a planar moving layer for 1 ⫽2⫽0.5, k⫽/c⫽1 cm⫺1, ␥⫽3, ⫽0.941: ⫽2 2 ⬎0 共slow waves兲, 䊏 䊏 䊏⫽22⬍0 共fast waves兲, h⫽10 cm;
⫽2
2⬎0 共slow waves兲, h⫽1 cm.
TECHNICAL PHYSICS LETTERS VOLUME 25, NUMBER 12 DECEMBER 1999
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h→⫹⬁ the relative influence of the boundaries becomes negligible and Eq.共1兲 yields the dispersion equation for sur- face waves at the planar interface between two media mov- ing relative to each other:3
共21⫹12兲共1⫹2兲⫽共2⫺1兲共1⫺1兲 2␥2
y 2. 共2兲 Figure 1 gives the propagation constant kz as a function
of the angle between k⬜ and V for surface waves at the moving layer (k⫽1 cm,⫽0.941, ␥⫽2.96, 1⫽2⫽0.5). Surface waves whose amplitude decays exponentially with increasing distance from the boundaries outside the layer may have two types inside the layer: slow surface waves for 2
2⬎0 and fast volume waves for 2
2⬍0. Unlike an isolated tangential velocity discontinuity where the surface waves could not propagate at angles smaller than some critical angle kbetween k⬜ and V共Ref. 3兲, for surface waves at a
layer the critical anglekcorresponds to the transition from
fast to slow waves. The largest critical angle is achieved for S⬇
冑
12, which for 1⫽2 gives h→⫹⬁. A decrease in the layer thickness leads to a reduction in the critical angle and for small h no fast volume waves appear.In addition to stable solutions, the equations共1兲 may also have solutions which increase in time or space and lead to the appearance of absolute and convective instabilities of the plasma layer. Analytic expressions for a plasma external me- dium may be obtained in limiting cases of high and low frequencies by substituting expressions for the permittivity of a stationary and moving plasma into Eq.共2兲. For real k we then have ⫽z⫺1关1⫾ikp2k⫺1␥⫺1共共k2⫹kp1 2  y 2兲/共k p1 2 ⫺2k2兲兲1/2兴, 共3兲
where kⰆkp1, kⰆkp2 and for kⰇkp1, kⰇkp2.
When kzⰆkp1, kzⰆkp2 and kⰆkp1, kⰆkp2, the solu-
tion may be expressed in the form ⫽关z共1⫹兲⫾i共2共1⫹兲共1⫺z 2⫹共1⫺2兲兲兲1/2兴 ⫻共z 2⫹ 2兲⫺1, 共4兲 where ⫽kp2/kp1, kp1 2 ⫽4e2n 1/mc2, kp2 2 ⫽4e2n 2/ m␥c2, n1and n2 are the electron concentrations outside and inside the layer, e is the charge, and m is the electron mass.
Note that in these cases, the maximum spatial growth rate of the oscillations is reached forz→0,y⫽. In this case, we obtain from Eq. 共4兲 Im ()⫽(1⫹␥⫺2)
⫻(1⫹)␥/2.
By finding complex solutions of Eq. 共1兲 in the form of frequency perturbations of the wave whose phase velocity is equal to the velocity of the layer in its direction of propaga- tion k⫽k0⫹k*, 兩k*兩Ⰶk0, k0⫽zkz; ph⫽(k/kz) ⬇ (k0/kz)⫽z, we obtain k *⫽ kp2 ␥
冉
1 2⫹ 12S⫺共1⫺1兲␥2y 2 /z2 共1S⫹12兲共1S⫺1⫹2兲冊
1/2 , 共5兲 where12⫽(1⫺1z 2)/ z 2, 2 2⫽(1⫺ z 2)/ z 2⫹k p2 2 /k 0 2. Thus, for resonant wave instability the constraints 1⬍0 or1⬎1 must be satisfied, i.e., the external medium must be dielectric.
Figure 2 gives the real and imaginary parts of k obtained by means of a numerical solution of Eqs. 共1兲 and 共2兲 for a fixed real propagation constant kz as a function of the angle
between k⬜ and V for surface waves at the tangential velocity discontinuity between two media and a moving plasma layer for kz⫽1 cm⫺1, n1⫽n2⫽1.4⫻1011cm⫺1,
⫽0.999,␥⫽22.4, and h⫽1 cm. In the ultrarelativistic case
the tangential velocity discontinuity is stable with respect to surface waves whose wave vector is parallel to the velocity of the medium but unstable with respect to waves propagat- ing at angles greater than some critical angle . For the plasma layer the symmetric and antisymmetric waves rela- tive to the y z plane have different critical angles from which they begin to grow. The range of angles where the flux is stable is determined by the smaller of these.
This type of surface waves may be used in relativistic plasma microwave electronics devices to generate electro- magnetic waves in directions other than the direction of mo- tion of the plasma layer.
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FIG. 2. Dependences of the real and imaginary parts of
k⫽/c on the angle between k⬜and V for waves at a tangential velocity discontinuity and at a moving plasma layer for ⫽0.999 (␥⫽22.4), kz⫽1 cm, n1⫽n2⫽1.4 ⫻1011
cm, h⫽1 cm: ⫽Im(k), 䊏 䊏 䊏⫽Re (k), planar
moving layer; ⫽Im (k), - - - ⫽Re(k), tangential ve- locity discontinuity.
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