Abstract
In this paper the authors have considered the mutual focusing/defocusing of a number of coaxial Gaussian electromagnetic beams in a singly ionized collisional plasma (initially in thermal equilibrium) and the ionosphere (with singly charged ions). Starting from the expression of the electron temperature in terms of the irradiance of the waves, expressions for the electron density and the dielectric function in the form have been derived; the power loss by electrons to heavy particles is assumed to be much larger than that due to thermal conduction. The dominant nonlinearity considered herein is the radial redistribution of the electron density on account of the radial dependence of the electric field of the waves and consequently of the electron temperature. Using this expression for the dielectric function, the coupled wave equations corresponding to different beams have been solved in the paraxial approximation, yielding a system of coupled secondorder differential equations for the beamwidths. The coupled equations for the widths of two beams have been solved numerically for some typical cases; the critical curves for the two beams have also been obtained. The effect of one beam on the critical curve and the dependence of the beamwidth on the distance of propagation of the other beam have been specifically considered. The results have been presented in the form of graphs for plasmas in thermal equilibrium and also for daytime midlatitude ionosphere at a height of . A discussion of the results is also presented.
The authors are grateful to Professor M.P. Verma for valuable discussion.
We also gratefully acknowledge the Department of Science and Technology, Government of India, for financial support.
I. INTRODUCTION
II. ANALYSIS
A. Radial distribution of electron density
B. Propagation of Gaussian electromagnetic beams in the plasma/ionosphere
C. Evaluation of in the collisional plasma (initial thermal equilibrium)
D. Evaluation of () for the ionospheric plasma
III. NUMERICAL RESULTS AND DISCUSSION
IV. CONCLUSIONS
Key Topics
 Ionospheric plasmas
 27.0
 Irradiance
 15.0
 Plasma electromagnetic waves
 13.0
 Plasma waves
 13.0
 Dielectric function
 11.0
Figures
Critical curve for a collisional plasma: Dependence of on axial irradiance of the first beam , when the axial irradiance of the second beam (a) 0.0; (b) 0.1; (c) 1.0; and (d) 5.0; the plasma and beam parameters are , , and . The case corresponds to a collisional plasma, which is in thermal equilibrium in absence of the beams.
Critical curve for a collisional plasma: Dependence of on axial irradiance of the first beam , when the axial irradiance of the second beam (a) 0.0; (b) 0.1; (c) 1.0; and (d) 5.0; the plasma and beam parameters are , , and . The case corresponds to a collisional plasma, which is in thermal equilibrium in absence of the beams.
Critical curve for a collisional plasma—another representation: Variation of the function of the first beam on the ratio of square of initial beamwidth of the two beam , when the axial irradiance of first beam (a) 0.1; (b) 1.0; (c) 2.0; (d) 5.0; (e) 10.0, and the axial irradiance of the second beam is . The case corresponds to a collisional plasma, which is in thermal equilibrium in absence of the beams.
Critical curve for a collisional plasma—another representation: Variation of the function of the first beam on the ratio of square of initial beamwidth of the two beam , when the axial irradiance of first beam (a) 0.1; (b) 1.0; (c) 2.0; (d) 5.0; (e) 10.0, and the axial irradiance of the second beam is . The case corresponds to a collisional plasma, which is in thermal equilibrium in absence of the beams.
Dependence of beamwidth parameter (solid lines) and (dashed lines) on distance of propagation . The curve for corresponds to and (a) 0.0; (b) 1.0; (c) 5.0, and curves for correspond to and (d) 0.0; (e) 1.0; 5.0 (g); the other parameters are , , , (when one beam is much weaker than the other). The case corresponds to a collisional plasma, which is in thermal equilibrium in absence of the beams.
Dependence of beamwidth parameter (solid lines) and (dashed lines) on distance of propagation . The curve for corresponds to and (a) 0.0; (b) 1.0; (c) 5.0, and curves for correspond to and (d) 0.0; (e) 1.0; 5.0 (g); the other parameters are , , , (when one beam is much weaker than the other). The case corresponds to a collisional plasma, which is in thermal equilibrium in absence of the beams.
Dependence of beamwidth parameter (solid lines) and (dashed lines) on distance of propagation . Different curves correspond to (a) and (c) , ; (b) and (d) ; other parameters are the same as for Fig. 3 (both beams have comparable axial irradiance). The case corresponds to a collisional plasma, which is in thermal equilibrium in absence of the beams.
Dependence of beamwidth parameter (solid lines) and (dashed lines) on distance of propagation . Different curves correspond to (a) and (c) , ; (b) and (d) ; other parameters are the same as for Fig. 3 (both beams have comparable axial irradiance). The case corresponds to a collisional plasma, which is in thermal equilibrium in absence of the beams.
Critical curve for ionosphere: Dependence of in midlatitude daytime ionospheric plasma (at height ) on axial irradiance of the first beam , when the axial irradiance of the second beam (a) 0.0; (b) 5.0; (c) 10.0; (d) 15.0; the beam parameters are , , .
Critical curve for ionosphere: Dependence of in midlatitude daytime ionospheric plasma (at height ) on axial irradiance of the first beam , when the axial irradiance of the second beam (a) 0.0; (b) 5.0; (c) 10.0; (d) 15.0; the beam parameters are , , .
Dependence of beamwidth parameter (solid lines) and (dashed lines) on distance of propagation . The curve for corresponds to , (a) 50.0; (b) 100.0 curves for correspond to , (c) 50.0; (d) 100.0. and the other parameters are the same as in Fig. 5.
Dependence of beamwidth parameter (solid lines) and (dashed lines) on distance of propagation . The curve for corresponds to , (a) 50.0; (b) 100.0 curves for correspond to , (c) 50.0; (d) 100.0. and the other parameters are the same as in Fig. 5.
Dependence of beamwidth parameter (solid lines) and (dashed lines) on distance of propagation in inhomogeneous collisional plasma . The curves for correspond to and (a) 10.0; (b) 0.0; curves for correspond to and (c) 10.0; (d) 0.0. The other parameters are , , and . The case corresponds to a collisional plasma, which is in thermal equilibrium in absence of the beams.
Dependence of beamwidth parameter (solid lines) and (dashed lines) on distance of propagation in inhomogeneous collisional plasma . The curves for correspond to and (a) 10.0; (b) 0.0; curves for correspond to and (c) 10.0; (d) 0.0. The other parameters are , , and . The case corresponds to a collisional plasma, which is in thermal equilibrium in absence of the beams.
Tables
Dependence of (in ) on height (in kilometers) for midlatitude daytime ionosphere.
Dependence of (in ) on height (in kilometers) for midlatitude daytime ionosphere.
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