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The Scattering Phase Shift Due to Bragg Resonance in One-Dimensional Fluctuation Reflectometry
Bedros B. Afeyan
Lawrence Livermore National Laboratory
Livermore, California, 94551
Albert E. Chou
Department of Electrical Engineering
University of California, Los Angeles
Los Angeles, California, 90024
Bruce I. Cohen
Lawrence Livermore National Laboratory
Livermore, California, 94551
An explicit integral representation is derived for the tangent of the phase shift due to one-dimensional (1-D) scattering of an S-polarized, O-mode, electromagnetic field from a localized wavepacket sitting atop an inhomogeneous plasma. A Green's function technique is used in the derivation together with the Born approximation. The integral representation is evaluated using asymptotic techniques and Bragg resonance is seen to be the dominant mechanism producing the phase shifts due to fluctuations whose wavelengths are short compared to the Airy length. By suitably normalizing the governing differential equation, we have identified the two dominant parameters that control the approximations in our analysis. These are 
and 
. The first is the maximum density fluctuation magnitude multiplied by the square of the dimensionless length scale that characterizes both the background plasma density profile (whose scalelength is L) and the incoming microwave field (whose vacuum wavenumber is 
): 
. The second is the fluctuation wavenumber normalized to and scaled by the similarly normalized Airy wavenumber: 
. The Born approximation is expected to be valid as long as < 1, and the Bragg resonance picture dominates as long as > 1.
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The Bragg Resonance Picture of One-Dimensional Fluctuation Reflectometry Beyond the Born Approximation
Albert E. Chou
Department of Electrical Engineering
University of California, Los Angeles
Los Angeles, California, 90024
Bedros B. Afeyan
Bruce I. Cohen
Lawrence Livermore National Laboratory
Livermore, California, 94551
By using HELM1D, a numerical, one-dimensional Helmholtz equation solver, we have studied the scattering phase shift of S-polarized O-modes in fluctuation reflectometry both within and beyond the Born approximation.
Comparisons are made between these numerical results and analytical expressions obtained through the use of the Born approximation. As the fluctuation amplitude increases, various peaks of the phase response evolve with distinct growth rates. This leads to steepening and distortion of the scattering phase shift due to a single wavenumber fluctuation. The simultaneous coexistence of
large amplitude fluctuations with nearby wavenumbers complicates the task of resolving the response at each wavenumber.
Overview of results
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