Do radio beams self-focus in the interstellar medium?

Summary: In a plasma, electromagnetic waves create a ponderomotive force that pushes electrons and ions out of the way. Thus, in an intense laser beam, electrons tends to move away from the areas where the beam is strongest. This changes the local refractive index of the plasma, focusing the beam more tightly and counteracting diffraction. At optical frequencies, this effect has been known for decades and has been used for laser-driven fusion, laser wakefield acceleration, high harmonic generation, and other applications. Thus, my question is: does the same self-focusing effect exist for radio waves and would it be significant in the interstellar medium?

Attempt at an answer: For an electromagnetic wave propagating in the $$z$$-direction, the electric field vector satisfies the scalar wave equation in cylindrical coordinates

$$\frac{\partial^2 E(r,z)}{\partial z^2} + \frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial}{\partial r}E(r,z)\right) + \frac{\omega^2 \varepsilon (r,z)}{c^2} E(r,z) = 0$$

where $$\varepsilon(r,z)$$ is the relative permittivity, which can be split into components $$\varepsilon(r,z) = \varepsilon_0(z) + \varepsilon_1(r,z)$$. If $$\varepsilon=1$$, we get an ordinary electromagnetic wave propagating along the $$z$$-axis; so the key question is how the nonlinearity in $$\varepsilon$$ created by the ponderomotive force affects the solutions.

Following Akhmanov et al., we can solve the wave equation using the first-order WKB approximation. The general result is

$$E(r,z) = A(r,z) \left(\frac{\varepsilon_0(0)}{\varepsilon_0(z)}\right)^{1/4} \exp\left( -\frac{i\omega}{c} \int^z \sqrt{\varepsilon_0(z)}\, dz \right)$$

where we have adopted the simplified notation of Sharma et al. for convenience. We introduce the eikonal function $$S(r,z)$$ such that

$$A(r,z) = A_0(r,z) \exp\left( -i\sqrt{\varepsilon_0(z)}\frac{\omega}{c}S(r,z) \right)$$

We will assume that the beam is initially Gaussian. Then in the paraxial approximation, one can show that

\begin{align} S(r,z) &= \frac{r^2}{2}\beta(z) + \phi(z) \\ (A(r,z))^2 &= \frac{E_{00}^2}{(f(z))^2} \exp\left( -\frac{r^2}{r_0^2 (f(z))^2} \right) \end{align}

where $$E_{00}$$ is the initial electric field strength and $$r_0$$ is the initial characteristic radius of the beam; $$f(z)$$ is the beam width parameter, which satisfies

$$\varepsilon_0(z) \frac{d^2f}{dz^2} = \left( \frac{c^2}{r_0^4 \omega^2} + \frac{\varepsilon_1(r,z)}{r^2} f^4 \right)$$

and $$\beta(z)$$ is the curvature of the wave front, which satisfies

$$\beta(z) = \frac{1}{f}\frac{df}{dz}$$

To find a specific solution, we now need the effective permittivity $$\varepsilon(r,z)$$. For a collisionless plasma under the influence of a ponderomotive force, we find that

$$\varepsilon = 1 - \Omega^2 \exp\left( -\frac{3}{4}\frac{m_e}{M}\alpha{EE}^* \right)$$

as given by Sodha et al. (p. 190). Here

\begin{align} \Omega &= \frac{\omega_p}{\omega} \\ \omega_p &= \sqrt{\frac{n_e e^2}{m_e \varepsilon_0}} \\ \alpha &= \frac{e^2 M}{6 k_B T_0 \omega^2 m_e^2} \end{align}

and the symbols are defined as follows: $$\omega_p$$ is the plasma frequency; $$\omega$$ is the frequency of the incident electromagnetic wave; $$n_e$$ is the electron density; $$e$$ is the electron charge; $$m_e$$ is the electron mass; $$\varepsilon_0$$ is the permittivity of free space; $$M$$ is the mass of an ion; $$k_B$$ is the Boltzmann constant; $$T_0$$ is the equilibrium plasma temperature; $${EE}^*$$ is the wave intensity represented as the mean square of the electric field, i.e. avg($$|E|^2$$). For the wave intensity, we plug in a Gaussian function of the form

$${EE}^* = \sqrt{\frac{\varepsilon_0(1)}{\varepsilon_0(f)}} \left( \frac{E_{00}^2}{f^2} \right) \exp\left( -\frac{r^2}{r_0^2 (f(z))^2} \right)$$

Defining

$$p= \left(\frac{3m_e}{4M}\right)\alpha \sqrt{\frac{\varepsilon_0(1)}{\varepsilon_0(f)}} \left( \frac{E_{00}^2}{f^2} \right)$$

and making the approximation $$\exp(-r^2/r_0^2 (f(z))^2) \approx 1-r^2/r_0^2 (f(z))^2$$, we find after a little bit of math that

\begin{align} \varepsilon_0(f(z)) &= 1-\Omega^2 e^{-p} \\ \varepsilon_1(r,f(z)) &= -\Omega^2 p e^{-p} \left( \frac{r^2}{r_0^2 (f(z))^2} \right) \\ \end{align}

We are now ready to put everything together. Plugging $$\varepsilon_0$$ and $$\varepsilon_1$$ into the differential equation for $$f(z)$$, we obtain

$$(1-\Omega^2 e^{-p}) \frac{d^2f}{dz^2} = \frac{c^2}{r_0^4 \omega^2} - \Omega^2 p e^{-p} \frac{(f(z))^2}{r_0^2}$$

On the right side, the positive term represents diffraction and the negative term represents self-focusing. We are interested in the boundary between a self-focusing beam and a divergent one, which occurs when these two terms cancel each other out. In this critical case, the beam width should be constant; thus, $$f(z)=1$$ and $$\frac{d^2 f}{dz^2} = 0$$. Furthermore, $$\varepsilon_0(z)$$ is constant and $$p$$ is a constant $$p_0 = \frac{3m_e}{4M} \alpha E_{00}^2$$. Making these substitutions, we find that the critical case is described by

\begin{align} 0 &= \frac{c^2}{r_0^4 \omega^2} - \Omega^2 p_0 e^{-p_0} \frac{1}{r_0^2} \\ \Rightarrow \frac{c^2}{r_0^4 \omega^2} &= \Omega^2 p_0 e^{-p_0} \frac{1}{r_0^2} \\ \Rightarrow r_0^2 &= \frac{c^2}{\omega_p^2} \frac{e^{p_0}}{p_0} \end{align}

(note that the frequency dependence is hidden inside $$p_0$$). This matches the results of Akhmanov et al. and other later researchers. Beams larger than this critical radius will self-focus, while smaller beams will diverge.

We can now plug some numbers in. For the Local Interstellar Cloud, $$T_0 \approx$$ 7000 K and $$n_e \approx$$ 0.05 cm$$^{-3}$$ (source). Let us choose $$\omega =$$ 1 MHz and $$E_{00} =$$ 5 V/m. Then $$p_0 = 0.911$$ (Wolfram Alpha); $$\omega_p =$$ 12.6 kHz (Wolfram Alpha); and we find a critical beam width $$r_0 \approx$$ 39 km. That implies a total beam power of somewhere around 150 megawatts.

So did I make an elementary mistake somewhere or do these equations actually describe a real, physical phenomenon? If everything above is correct, one would expect to see self-focusing beams of radio waves in the ionosphere and various other astrophysical environments. Has anyone looked for such effects? (This paper is relevant, but doesn't mention any experimental evidence.) Furthermore, this would appear to have some rather interesting applications for interstellar travel and communication; notably, it would seem that one could relatively easily build a spacecraft with a large superconducting antenna and propel it to another star system using radiation pressure.