Brillouin-Enhanced Four-Wave Mixing and Phase Conjugation

Detailed experimental investigations of Brillouin-enhanced four-wave mixing(BEFWM) and phase conjugation of microwaves in an unmagnetized plasma have been performed. Through the use of plasma probes and energy analyzers together with a millimeter-wave interferometer/scattering system, both the transient and steady-state responses of the plasma and the phase conjugate wave have been investigated. Low power, low density operation is shown to agree well with two-fluid plasma theory. Strong suppression of the BEFWM interaction is observed at an electron density n_e=n_c/4, indicative of the 2w_pe instability. High power, high density operation is seen to deviate from simple theoretical predictions; slight amounts of plasma pushed out of the interaction region by the ponderomotive force of the high power pump waves are observed to disrupt the relative phases of the electromagnetic waves involved in the BEFWM interaction.

BEFWM Theory

Fig. 1 Four-wave mixing geometry

Four-wave mixing (FWM ), in its simplest form, can be modeled as a pair of simultaneous three-wave mixing processes. In the first(optical mixing) process, a strong pump wave mixes with a weak signal wave in a nonlinear medium (such as a plasma) to generate a density modulation or "grating" in the medium. In the second (scattering) process, another strong pump wave scatters off this grating to generate a fourth wave. If the two pump waves are antiparallel, the fourth wave is then phase conjugate to the signal wave, i.e. its wavefronts coincide everywhere with the signal wave and are counterpropagating.

Experimental Measurements

A 200cm long, 75cm diameter stainless steel vacuum vessel in which low density unmagnetized H₃+ ion filament discharge plasmas are produced, is employed for experimental BEFWM studies. Due to the 38 degree tilt angle between the pump and signal waves, both a large-k and a small-k grating

are formed in the plasma. The conjugate wave thus results from the scattering of E₂ off the large-k grating and of E₁off the small-k grating. And due to the rather large wavelengths utilized, considerable diffraction and beam spreading occurs within the chamber. A forth-order high pass filter section was employed to filter out the undesired pump and signal wave power in order to accurately measure the conjugate power collected by the signal horn.

Fig. 2 Schematic of the plasma chamber, depicting the size of the Gaussian beams formed by pump and signal horns assuming a plasma density n₀ = 0.1n_c.

Phase measurements, in which calibrated phase delays were placed along various points in the signal and conjugate wave paths, show that out put wave is indeed phase conjugate to the input signal wave.

Fig. 3 Output conjugate wave phase measurements, taken with a variable phase delay inserted in the signal wave path alone, in the conjugate wave path alone, and in the path shared by both signal and conjugate waves.

Figure 4 displays the scaling of the conjugate wave signal with plasma density, taken with power levels of P₁=10 kW, P₂=7.5 kW, and Ps=1.2 kW. at low densities, the signal is observed to indeed grow linearly with increasing plasma density. The conjugate wave amplitude is seen to drop sharply near the quarter-critical layer, disrupted possibly by the large amplitude ion fluctuations often observed in presence of the 2w_pe instability.

Fig. 4 Conjugate wave amplitude as a function of plasma density.

Figure 5 illustrates the effect that increasing pump power has upon the conjugate wave. The "oscillations" have been observed to grow in size and frequency as the pump and plasma density are increased.

Fig. 5 Conjugate wave signals at a plasma density of 0.20n_c

Independent evidence of ponderomotive whole-beam self-focusing is found through the use of the millimeter-wave interferometer system. In Fig. 6 are line-integrated plasma density waveforms, first of a low density plasma discharge followed by a higher density plasma discharge. Simultaneous measurements using a planar Langmuir probe, moveable along the chamber axis, confirm the interferometer results.

Fig. 6 Line-integrated plasma density waveforms obtained by the millimeter-wave interferometer, of plasmas irradiated with 50us pulses of high power microwaves.

Measurements of the conjugate wave power Pc as a function of signal wave power Ps provides further proof that the temporal distortions observed at high power levels arise from the ponderomotive whole-beam self-focusing instability, and not from, say, ion wave saturation which is dependent upon both the pump wave and signal wave power levels.

Fig. 7 Conjugate power Pc as a function of signal power Ps, at time t=15us, and t=45us, with a plasma density of 0.20n_c

Conclusions:

Brillouin-enhanced four-wave mixing in an unmagnetized plasma has been clearly demonstrated. The detailed measurements have proven the validity, under low gain(i.e. low power and low density) conditions, of two fluid plasma theory. High power, high density operation is seen to deviate strongly from theory; evidence has been gathered to support the hypothesis that much of the deviation can be attributed to ponderomotive whole-beam self focusing of the high power pump waves.