Second-Harmonic Generation of Electron-Bernstein Waves in an Inhomogeneous Plasma

In the injection of electron-Bernstein waves (EBW) into a plasma, proposed for plasma heating and current drive in over-dense plasma, conversion of the fundamental to its second harmonic is predicted analytically and observed in computations. The mechanism is traced to the existence of locations where one can have both wave number and frequency matching between the fundamental and its harmonic. Further, at such locations, the second harmonic commonly has minimal group velocity, and this allows the amplitude of the second harmonic to build to values exceeding that of the fundamental at power levels less than anticipated in experiments. The second-harmonic power can then be deposited at half-harmonic resonances of the original wave, often far from the desired location of energy deposition. Estimates for the power at which this is significant are given.

Figure 1

Dispersion curves of EBWs for ${\omega }_e/{\Omega }_e=3.5$.

Figure 2

(a) Simulation model and the plasma density profile, (b) normalized frequencies with the resonant coupling layer indicated.

Figure 3

EBW dispersion curves $k(x,\omega )$ (solid line), $2k(x,\omega )$ (light dashed line), and $k(x,2\omega )$ (heavy dashed line). Circles are the measured values of $k$ from the simulation for the fundamental. Squares the measure values of $k$ from the simulation for the harmonic.

Figure 4

Spatial distribution of the electrostatic fields ($E_x$) at the driving frequency [(a) and (b)] and its second harmonic [(c) and (d)] at $t=245/{\omega }_0$ and $454/{\omega }_0$. The location of the resonant coupling is indicated in (a) by the dashed line.

Figure 5

(a) Logarithm of $|E_x|$ vs $x$ at the driving frequency from different incident fields, $E_0=2\times {10}^4$, $4\times {10}^4$, $5\times {10}^4$, $6\times {10}^4$, and $8\times {10}^4\mathrm{V}/\mathrm{m}$ (from bottom to top). (b)–(c) show the amplitude of the second EBW vs the square of the amplitude of the fundamental EBW from the $\delta f$ simulations (circles) at $x=0.1456\mathrm{m}$, 0.1473 m, and 0.1489 m, respectively.

Figure 6

Dispersion curves for pump EBW (circle) and second-harmonic EBW (diamond). The dotted line indicates the wave number twice the fundamental wave, and the cyclotron resonance layers are marked.

Figure 7

Spatial distribution of the electrostatic fields ($E_x$) at the driving frequency [(a)–(c)] and its second harmonic [(d)–(f)] at $t=252/{\omega }_0$, $532/{\omega }_0$, and $1405/{\omega }_0$, respectively. The cyclotron resonance layers $2{\omega }_0=3{\Omega }_e$ and ${\omega }_0={\Omega }_e$ are indicated in (a) and (d) by the dashed and dotted lines, respectively.

Figure 8

Time history of $u_x^2=(V_x/V_t{)}^2$ at (a) $t=252/{\omega }_0$, (b) $532/{\omega }_0$, and (c) $1405/{\omega }_0$.