The development of a cold plasma theory for a nonuniform density profile has been hindered because the steady-state plasma response and. The calculated reflectance of the plasma-loaded waveguide showed one peak when the hybrid resonance was in the center of the plate and another at w/w = 1. The effects of the glass tube surrounding the plasma were also explicitly included in the calculation.
Baldwin [20] and Baldwin and Ignat [21] using the electrostatic approximation arrived at analogous conditions for the validity of the cold plasma theory in the case without a magnetic field.
REFLECTION
REGION EINC ~o
8 tNC
THEORETICAL
TRANSMISSION REGION
These results will be used in the treatment of the bar line problem in Section 2.4. Outside the plasma column, the field consists of the incident plane wave plus some scattered waves as shown in Fig. The linearity of the field equations and the boundary condition at the origin determine the solutions of the field inside the plasma up to a multiplicative constant.
The set of equation (2.9) completes the solution of the plane wave problem given any set of solutions to Eqns. 2.3) which are regular in origin.
STRIP LINE IMAGES
COLUMN
COLUMN UPPER CONDUCTING
PLANE- LOCATJON- OBSERVATION
Using the linearity of the plasma response, the problem can be expressed in terms of plane wave coefficients. This can be achieved by noting that the image column fields definition, . As long as the plasma response is linear, these two complex coefficients completely determine the properties of the plasma column at the bar line.
Direct experimental verification of the fields and plasma disturbances predicted from the solutions of Eqns. The results in this section are calculated for parameter values such that ~90% of the plasma response appears in the m·+l mode. Therefore, the solutions below are shown only for the m-+1 mode. The effect of the plasma on the internal electromagnetic fields is illustrated by graphs of the magnitudes of the electric field components IEr(r)l and IE.
The most striking feature of the plasma response is the existence of a strong, narrow resonance zone near which lies the base. There is no evidence for the vanishing region predicted by theory for a homogeneous plasma visible in the calculations. The effect of changing the value of damping is illustrated by the graphs of IE I r shown in Fig.
1 • Since the scattering properties of the column are completely determined by the solutions at r s r, the scattering will also show a trough. The resonance in the absorption is a reflection of the shape predicted for the damping from Eqs.
Another set of experimental results can also be interpreted based on the existence of a hybrid layer. The frequency of the heating pulse was such that it could excite the hybrid layer in the plasma and was used at a fixed afterradiation time and magnetic field. The center frequency of the radiometer and, assuming the explanation developed above is correct, the radius it detects varied with the results shown in Figure 2.
ARGON
RADIUS
An important feature of the cold plasma results presented in chapter 3 is the strong plasma excitation in the hybrid layer. The major result of including finite temperature effects in a magnetoplasma is the introduction of Bernstein [16] modes. There is thus no guarantee that the solutions of the low-temperature equations are small perturbations of zero-.
In the following sections, both the cold and low temperature plasma equations will be replaced by simplified wave equation models for E. The explicit coupling formula for r = 0 will be useful for comparison with the results of the warm plasma of Section 4.3. Solutions on the high-density side of the coupling region are shown to the left of .
This interpretation decouples the modes on the low density side of the hybrid layer in the solution sets as given, and the linear combinations B. In this case, the inclusion of finite temperature effects will not change the cold plasma field solutions outside the region. the hybrid layer and the cold plasma theory should be sufficient to predict the scattering properties of the plasma. 3 will be negligible, as will be discussed below, there is an experimentally observable consequence of the presence of an outgoing Bernstein wave on the hybrid layer.
3 is negligible within that range and the scattering properties calculated from cold plasma theory should be valid. These conditions will be investigated based on the results presented in Chapter 3 for typical experimental parameters. N is required in the linearization of the currents in Maxwell's equations and is thus a prerequisite for the generation of negligible harmonics.
1 ~sec and 100 ~sec and that the temperature of the hybrid layer decreased on a time scale of 1/2 msec after the end of the heating pulse.
POWER ABSORPTION
6. dispersion coefficients) are indistinguishable for all parameter sets which give the same value of the scaling function. Recalling the discussion in section 2.2, the plasma response is completely described by the dielectric tensor components K~(r) and Kx(r) , which can be written in the form The plasma response will not be distinguishable for two different parameter sets if the functions K~(r) and Kx(r) are identical for both parameter sets.
Thus, there is no useful exact scaling relation, but a useful approximate scaling relation exists for low densities. Since
The existence of the collision-free limit, as discussed in Chapter 3, suggests that the first of Eq. 5.2) should be more important than the second. A quantitative analytical assessment of the validity of the sealing is difficult since it requires some estimation of the sensitivity of the solutions at r = r to the errors in the coefficients of Eq. 2.3) caused by the approximate nature of the scale. As an alternative, the effectiveness of the scaling relationship was investigated computationally.
POWER REFLECTION
For these reasons, the remaining discussion of the low-density case except for a set of calculated curves for direct comparison with experimental results will be. The left curves represent the reflection from a plasma column without a glass envelope and the right curves represent The effects of the glass tube on scattering are not simple as the fields scattered by the plasma modify the dielectric response in the glass and vice versa.
The strength of the glass effects suggests that they should be included to make quantitative comparisons between them. The reflection coefficients measured in these studies show two peaks, one near (w /w) and another near w /w = 1.0, which is. qualitatively different from the theoretical results obtained in the present study either with or without the glass tube. Thus, these curves represent slowly varying values for the second scaling parameter used in Eq. 5.2) and the labels actually give the value of this parameter at w lw = 1.
5 x 10-3• The main effects appear to be the increasing magnitude of the scattering and the shift of the peaks towards lower values of w lw. The theoretical results to be presented in this section have been calculated using the parabolic profile as a reasonable approximation in the afterglow when ambipolar effects should be significant. The central density for each trial curve was calculated from the observed value (w lw) for absorption assuming that.
The value of
ABSORPTION COEFFICIENT
REFLECTION COEFFICIENT
The reflection coefficient in a waveguide is also affected by the presence of the holes in the side walls. The high density minimum in the transmission corresponds to strong excitation of the hybrid layer in the m = -1 state. This report has presented the results of a theoretical study of the nature of the upper hybrid resonance which occurs in the model of an inhomogeneous cold magnetoplasma.
Since it is difficult to directly measure the internal state of the excited plasma, a consideration-. In addition, the consequences of neglecting finite tern perature effects were inherent in the cold plasma model. This treatment also demonstrates that the onset of enhanced reflection or scattering is not a reliable indicator of the maximum upper hybrid frequency due to the effects of the glass tube and strip line.
The existence of the hybrid resonant layer also suggests the possibility of nonlinear behavior at low power. Although further adjustments in the cold plasma calculation may not provide much greater understanding of the physical nature of the hybrid layer in a plasma, it may still be worth extending this. The current computational algorithm could be extended to explicitly include the effects of the Bernstein modes.
This extended calculation would include a numerical solution of the fourth-order warm plasma wave equation in the coupling region near the hybrid layer and a WKB. The possibility of strong plasma excitation in the hybrid layer suggests that it should be useful for nonlinear studies. Strong excitation of the hybrid layer can also be valuable in producing localized hot regions in a plasma.
The treatment of the plane wave problem presented in Section 2.3 results in Eqs. 2.9) for the scattering coefficient of plane waves S.
