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Threshold for electron trapping nonlinearity in Langmuir waves
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Image of FIG. 1.
FIG. 1.

Trapped electron fraction due to transverse side loss for a 2D plane (blue) and 3D cylinder (red) region. The dashed curves are the appropriate early and late time limits. See Eqs. (6) through (12).

Image of FIG. 2.
FIG. 2.

(a) Amplitude of mode of , scaled to linear response, vs. time for 1D LOKI case 1D.35b and four different resolutions: (black), (64, 768) (green), (32, 384) (blue), and (64, 384) (red). (b) vs. dynamic bounce number from Eq. (1) using from 1D LOKI runs in Table I (see table for curve meanings). (c) Panel (b) for expanded domain. The black curve is the only run in (a) that appears in (b) and (c).

Image of FIG. 3.
FIG. 3.

Amplitude of mode of for 2D LOKI runs with transverse driver profiles h(y) with various . Run parameters and curve meanings are given in Table II. Black curve is 1D run 1D.35c.

Image of FIG. 4.
FIG. 4.

Departure from linear response for 2D LOKI runs, scaled to the same quantity from the 1D run 1D.35c. The colored curves are taken at times when the dynamic bounce number has reached the value indicated by the colored text. is found from linear response, using Eq. (20).

Image of FIG. 5.
FIG. 5.

Trapped fraction due to collisions vs. unitless time defined in Eq. (27). Solid curves are exact results from Eq. (29). , (4,1), and (4,4) for black, red, and blue, respectively. Dashed red and blue curves are approximate results for from Eq. (A19), which depend only on and not . Green dashed curve is , the approximate form neglecting the term proportional to in the denominator. and not unity due to electrons not initially in the fundamental mode.

Image of FIG. 6.
FIG. 6.

Relative error between the exact and approximate results from Eqs. (29) and (30), respectively, for collisional ; see text for details. The green curves are .

Image of FIG. 7.
FIG. 7.

Wavelength in nm for SRBS light (increments of 50 nm), for a pump wavelength . Black solid: phase-matched with a natural Langmuir wave, satisfying the dispersion relation Eq. (36) with . Red dash: for the maximum local SRBS spatial gain rate.

Image of FIG. 8.
FIG. 8.

Langmuir wave phase-matched for SRBS with different choices, for a pump wavelength . Black solid: phase-matched with a natural Langmuir wave, as in black solid curves of Fig. 7. Red dash: ('s fixed, k's vary). Blue dot: black-solid contour from Fig. 7.

Image of FIG. 9.
FIG. 9.

Threshold Langmuir wave amplitude for side loss, , for .

Image of FIG. 10.
FIG. 10.

Threshold Langmuir wave amplitude for collisions, . LW and are for phase-matched SRBS with a natural LW (black solid curves of Fig.7), and pump wavelength . Black solid: . Red dash: . From right to left, curves are for .

Image of FIG. 11.
FIG. 11.

Ratio from Figs. 9 and 10 for .

Image of FIG. 12.
FIG. 12.

Critical amplitude from Eq. (35) for . The side loss detrapping rate is faster than the collisional one for . Blue curves are for .

Image of FIG. 13.
FIG. 13.

Critical intensity from Eq. (40) in relating laser intensity and reflectivity R to LW amplitude: .

Image of FIG. 14.
FIG. 14.

Threshold reflectivity from Eq. (41) for trapping to overcome side loss for .

Image of FIG. 15.
FIG. 15.

Black: angle separating small- from large-angle scattering from Eq. (A29). Red: Ratio from Eq. (A33), with the value marked.


Generic image for table
Table I.

1D LOKI runs with no transverse driver profile h(y). . is found using .

Generic image for table
Table II.

2D LOKI runs with transverse driver profile h(y). All runs have , and , the same as run 1D.35c.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Threshold for electron trapping nonlinearity in Langmuir waves