Wave Properties in Cold Unmagnetized Plasma #

Before we can get to the wave properties, we first need to be specific about what we mean by a “cold” plasma. In the fluid formulation, we have a few central equations:

• Conservation of matter \[\pdv{n}{t} + \div (n \vec v) = 0\]

• Conservation of momentum \[\pdv{}{t} (m n \vec v) + \div (m n \vec v \vec v) = \sum \text{Force densities}\]

• Energy conservation \[\pdv{e}{t} + \div (e \vec v) = \sum \text{Energy sources}\]

For plasma waves, we are generally interested in the momentum equation. In a plasma, the source of the field dynamics is \( (\rho_c, \vec j) \)

\[\div \vec E = \rho_c / \epsilon_0\] \[\curl \vec B = \mu_0 \vec j + c^{-2} \partial_t \vec E\] \[\curl \vec E = - \partial _t \vec B\]

and the source of currents is the electromagnetic field

\[\vec j = \sum _\alpha n_\alpha q_\alpha \vec v_\alpha\] \[\pdv{\vec v_\alpha}{t} = \vec F = q _\alpha(\vec E + v_\alpha \cross \vec B)\]

We can write out the full momentum equation (with scalar pressure \( p \), antisymmetric pressure tensor \( \vec \Pi \), and collision operator \( \vec R \)):

\[\partial_t (m n \vec v) + \div (m n \vec v \vec v) = - \grad p - \div \vec \Pi + q n (\vec E + \vec v \cross \vec B) + \vec R\] We can also use index notation to make the tensor operations clearer. Repeated indices are summed together. \[\partial _i \equiv \pdv{}{x_i}\] \[\partial_i x_i \equiv \sum_i \partial_i x_i = \div \vec x\]

Writing out the momentum equation in index notation: \[\partial_t(n m v_i) + \partial_j (m n v_i v_j) = - \partial_i p - \partial_j \Pi _{ij} + q n (E_i + \epsilon_{ijk} v_j B_k) + R_i\]

There are a few types of forces that show up on the right-hand-side

• Thermal forces (\( p = n k T \))
  • \( - \grad p \): Pressure force
  • \( - \div \vec \Pi \): Viscous forces
• Body forces
  • \( q n (\vec E + \vec v \cross \vec B) \)
• Friction forces
  • \( \vec R \): Momentum transfer between species or other systems from friction When we look at a system where the temperature is low \( T \rightarrow 0 \), then the pressure is small \( p \rightarrow 0 \). What we mean by “cold” plasma is that we’re going to neglect pressure forces.

The cold plasma equation of motion for species \( \alpha \) is: \[\pdv{}{t} (m_\alpha n_\alpha \vec v_\alpha) + \div (m_\alpha n_\alpha \vec v_\alpha \vec v_\alpha) = q_\alpha n_\alpha (\vec E + \vec v_\alpha \cross \vec B)\] We can simplify using the continuity equation \[\pdv{}{t} (m_\alpha n_\alpha) + \div (m_\alpha n_\alpha \vec v_\alpha) = 0\] Moving back to index notation, we can simplify the mass continuity equation using the product rule \[\partial_t (mn) + \partial_j (mn v_j) = 0\] \[\partial_t (m n v_i) = v_i \partial_t (nm) + n m \partial_t v_i \] We can also use the product rule on the momentum flux \[\partial_j (m n v_i v_j) = v_i \partial_j (m n v_j) + m n v_j \partial_j (v_i)\] Collecting the \( v_i \) terms, we get a simplified equation of motion for the cold plasma \[n_\alpha m_\alpha \dv{\vec v_\alpha}{t} = q_\alpha n_\alpha (\vec E + \vec v_\alpha \cross \vec B)\] where we’re using the total derivative \[\dv{\vec v_\alpha}{t} \equiv \pdv{\vec v_\alpha}{t} + (\vec v_\alpha + \grad) \vec v_\alpha\]

Linearization of Equation of Motion #

If we encounter a non-linear differential equation, our technique of applying a Fourier transform to obtain the dispersion relation won’t work so well because we’ll get convolutions in the Fourier transforms. This means that we can’t solve non-linear differential equations with the same spectral method.

But, if we look at waves/perturbations of small amplitude, then linear dynamics are dominant and we can “linearize” the equation of motion to obtain an equation we can easily solve. If we only consider small amplitude waves, the non-linearity does not come into effect.

When we linearize an equation about an equilibrium, we are breaking up the equation into a hierarchy. We start by considering the Taylor expansion of our quantity of interest about some equilibrium \( f_0 \)

If we choose to expand about an equilibrium, then the zero-order component is static and we can ignore it. One of the most useful and simple equilibria is a homogeneous, flat, invariant plasma, meaning the plasma is characterized by a zero-order density that is constant, and is not moving.

\[n_0(\vec x, t) = n_0\] \[\vec v_0 (\vec x, t) = 0\]

We’ll also take the background electric field to be zero, because the background charge density is zero \[\vec E_0 ( \vec x, t) = 0\] And we’ll take the background magnetic field to be uniform and constant \[\vec B_0(\vec x, t) = \vec B_0 = \text{const.}\]

Of course, the really interesting stuff happens when we have an inhomogeneous plasma with pressure and density gradients, so we will be discussing the inhomogeneous case, but first we need to set up our foundation with the zero-order situation.

Our homogeneous expansion will be

\[n(\vec x, t) = n_0 + n_1 + \ldots\] \[\vec v(\vec x, t) = \vec v_1 + \ldots\] \[\vec E = \vec E_1 + \ldots\] \[\vec B = \vec B_0 + \vec B_1 + \ldots\] To obtain the first-order equation, we substitute our linearized quantities and ignore any products of first-order quantities. \[\partial _t (n_0 + n_1) + \div ((n_0 + n_1) \vec v_1) = 0\] The background density is constant in both space and time, so \[\partial_t n_1 + n_0 \div \vec v_1 = 0\] Doing the same thing with the momentum equation, \[m_\alpha\dv{\vec v_\alpha}{t} = q_\alpha(\vec E + \vec v_\alpha \cross \vec B)\] \[\dv{v}{t} = \pdv{\vec v}{t} + (\vec v \cdot \grad) \vec v\] In the total derivative, only the partial time derivative will survive to first order, because the other terms will be second-order. \[\vec v_\alpha \cross \vec B = \vec v_{1, \alpha} \cross (\vec B_0 + \vec B_1) \rightarrow \vec v_{1, \alpha} \cross \vec B_0\] \[\rightarrow m_\alpha \pdv{\vec v_{\alpha, 1}}{t} = q_\alpha(\vec E_1 + \vec v_{1, \alpha} \cross \vec B_0)\] Note that the first-order magnetic field is not part of the first-order linear dynamics

• \( E_1 \) has linear acceleration
• \( B_0 \) has linear acceleration So the electric and magnetic fields have different positions in our hierarchy of linearization. \[\partial _t n_{\alpha, 1} + n_{\alpha, 0} \div \vec v_{\alpha, 1} = 0\] \[\partial_t \vec v_{\alpha, 1} = \frac{q_\alpha}{m_\alpha} (\vec E_1 + \vec v_{\alpha, 1} \cross \vec B_0)\]

For a cold plasma, the linearized equation of motion is the same as the equation of motion for a single particle. Without pressure, all particles in the cloud of density \( n_\alpha \) will respond identically to the forces.

The initial velocity \( v_{\alpha, 1}(t = 0) \) is important, because the direction relative to \( \vec B_0 \) determines whether the plasma is magnetized.

Those are the plasma equations taken care of, but we also need the Maxwell equations. Lucky for us, Faraday’s law and Ampere’s law are already linear equations \[\curl \vec E = - \pdv{\vec B}{t}\] \[\curl \vec B = \mu_0 \vec j + c^{-2} \pdv{\vec E}{t}\] The only linearization we need to do is to consider only the current due to linear response in Ampere’s law.

We can combine Ampere’s law and Faraday’s law to get a second-order wave equation: \[\curl (\text{Faraday}) \rightarrow \curl \curl \vec E = - \pdv{}{t} \curl \vec B\] \[\pdv{}{t} (\text{Ampere}) \rightarrow - \pdv{}{t} \curl \vec B = - \mu_0 \pdv{}{t} \vec j - c^{-2} \pdv{^2 \vec E}{t^2}\] If we combine the two, we get \[\frac{1}{c^2} \pdv{^2 \vec E}{t^2} + \curl \curl \vec E = - \mu_0 \pdv{\vec j}{t}\] This is called the Helmholtz wave equation, when the space operator for the wave equation is given by two curls. Using the vector identiy \[\curl \curl \vec E = \grad (\div \vec E) - \grad ^2 \vec E\] we can re-write the Helmholtz wave equation \[\frac{1}{c^2} \pdv{^2 \vec E}{t^2} - \grad ^2 \vec E = - \mu_0 \pdv{\vec j}{t} - \epsilon_0 \grad \rho_c\] Alternatively, we can reverse the process with Faraday and Ampere to get a second wave equation for the magnetic field: \[\frac{1}{c^2} \pdv{^2 \vec B}{t^2} + \curl \curl \vec B = \mu_0 \curl \vec j\] Suppose we have free-space propagating waves with \( \vec j = 0 \) and \( \rho = 0 \), then we get \[\pdv{^2 \vec E}{t^2} - c^2 \grad ^2 \vec E = 0\] \[\pdv{^2 \vec B}{t^2} - c^2 \grad ^2 \vec B = 0\] So we expect that in free space, both \( \vec E \) and \( \vec B \) will propagate at speed \( c \). Also in free space, \[\div \vec E = 0 \qquad \div \vec B = 0\] \[\rightarrow \vec k \cdot \vec E = 0 \qquad \vec k \cdot \vec B = 0\] such that \( \vec E \) and \( \vec B \) must both be perpendicular to the direction of propagation; we will have transverse wave solutions.

In an unmagnetized plasma, \[\pdv{\vec v_{\alpha, 1}}{t} = \frac{q_\alpha}{m_\alpha} \vec E_1\] \[\vec j = \sum_\alpha q_\alpha n_\alpha \vec v_\alpha\] \[\rightarrow \pdv{\vec j}{t} = \sum_\alpha q_\alpha n_{0, \alpha} \pdv{\vec v_{1, \alpha}}{t}\] \[=\sum_\alpha \frac{q_\alpha ^2 n_{0, \alpha}}{m_\alpha} \vec E_1\] \[\pdv{\vec j}{t} = \left(\sum_\alpha \omega_{p, \alpha} ^2 \right) \epsilon_0 \vec E_1\] If we define a single combined plasma frequency \[\omega_p ^2 = \sum_\alpha \omega_{p, \alpha} ^2\]

\[\pdv{\vec j}{t} = \omega_p ^2 \epsilon_0 \vec E_1\] Plugging this back into the Helmholtz equation: \[\rightarrow \frac{1}{c^2} \pdv{^2 \vec E}{t^2} + \curl \curl \vec E = - \mu_0 \omega_p ^2 \epsilon_0 \vec E\] \[\pdv{^2 \vec E}{t^2} + c^2 \curl \curl \vec E + \omega_p ^2 \vec E = 0\] So we have a wave equation with a source term \( \omega_p ^2 \vec E \)

Transverse Waves #

If we look at the space propagation equation, \[\pdv{^2 \vec E}{t^2} - c^2 \grad ^2 \vec E = 0\] our solutions will be traveling plane waves \[\vec E = \vec E_0 e^{i (\vec k \cdot \vec x - \omega t)} \qquad - \omega ^2 + c^2 k^2 = 0\] Without loss of generality, if the wave is traveling in the \( z \) direction \( \vec k = k \hat z \), then \( \hat E_z = 0 \). Each of the other components can be written in terms of a complex phase \( \theta \) \[E_{0, x} = E_{00, x} e^{i \theta_x}\] \[E_{0, y} = E_{00, y} e^{i \theta_y}\] \[\vec E = E_x \hat x + E_y \hat y\] \[E_x = E_{00, x} e^{i (kz - \omega t + \theta_x)}\] \[E_y = E_{00, y} e^{i (kz - \omega t + \theta_y)}\] The wave equation in free space admits waves whose transverse components allow for arbitrarily different phase shifts. Plane wave solutions will fall into one of three categories:

• Linear polarization If one of the components is always zero, e.g. \( E_y = 0 \), then the field will oscillate along a single (linear) direction.
• Circular polarization We could also have \( E_{0, x} = E_{0, y} \). For example, a wave in which the phases are \[\theta_x = 0 \qquad \theta_y = - \pi / 2\] is a left-handed circularly polarized wave. Similarly \( \theta_x = 0 \) \( \theta_y = \pi/2 \) is a right-handed circularly polarized wave
• Elliptical polarization The general case we’ll run into is \( \theta_x, \theta_y \neq 0, \pi/2 \), \( \theta_x \neq \theta_y \), and \( E_{0, x} \neq E_{0, y} \). Then we’ll have an elliptical polarization diagram that spins and has a changing amplitude as it rotates.

From Faraday’s law: \[\vec B_1 = \frac{\vec k}{\omega} \cross \vec E\] If \( \omega \) is purely real, then \( \vec B_1 \) is perpendicular to \( \vec E \) and the two are completely in phase. However, if \( \omega \) is complex then they are not in phase and you’ll have a phase shift in the magnetic field relative to the electric field.

Coherent and Incoherent Polarization #

In general, the solution to \[\pdv{^2 \vec E}{t^2} = c^2 \grad ^2 \vec E\] is going to be a sum of a whole bunch of different frequency components \[\vec E = \int \vec A(\vec k) e^{i (\vec k \cdot \vec x - \omega t)} \dd k\] If \( \vec E \) has a single wavelength, then the wavefield is called coherent. But when a signal has many frequencies, as is the case with most light sources, then coherency falls apart quickly and we have incoherent polarization. Most light sources have no coherent polarization, and even if we pass light with some bandwidth through a polarizer it will only remain coherent for a limited time called the decoherence time, which is related to the frequency bandwidth \[\tau \sim 1 / \Delta f\] Lasers remain coherent for a long time because they have a very small frequency bandwidth.

Refraction #

We’re going to be talking about electromagnetic waves in media, so we’re going to need to go over a whole bunch of optics.

Refraction is what happens when a wave changes direction, generally as a result of inhomogeneity in the medium. The simplest material interaction is when light impinges on a surface.

We imagine a wave \( \vec k \) impinging on a material boundary (like the surface of water) with angle \( \theta_1 \). We expect some component to continue through the interface in the next medium, making angle \( \theta_2 \) with the interface.

Energy conservation demands that the frequency be continuous across the boundary \[\omega_1 = \omega_2\] If we let our boundary normal be in the \( y \) direction, then conservation of momentum in the tangential direction demands that \[k_{1, x} = k_{2, x}\] \[k_1 \sin \theta_1 = k_2 \sin \theta_2\] \[\frac{c k_1}{\omega_1} \sin \theta_1 = \frac{c k_2}{\omega_2} \sin \theta_2\] We define the index of refraction to get Snell’s law \[n \equiv \frac{ck}{\omega}\] \[\rightarrow n_1 \sin \theta_1 = n_2 \sin \theta_2\]

The index of refraction can also be written as the speed of propagation divided by the phase velocity \[n = \frac{c}{v_p} = \frac{\text{velocity in vacuum}}{\text{velocity in medium}}\]

Wave Properties in Unmagnetized Plasma #

Finally we get to the real deal. We take our equation for the time change of current density, and perform a Fourier transform \[\pdv{\vec j}{t} = \epsilon_0 \omega_p ^2 \vec E\] \[- i \omega \hat \vec j = \epsilon_0 \omega_p ^2 \hat \vec E\] \[\hat \vec j = i \frac{\epsilon_0 \omega_p ^2}{\omega} \vec E\] When we get a relationship between current and electric field, we’ve found a conductivity. \[\hat \sigma = i \frac{\epsilon_0 \omega_p ^2}{\omega}\] In a cold unmagnetized plasma, the conductivity is scalar and purely imaginary. From the definition of the general dielectric constant \[\epsilon = \epsilon_0 + i \frac{\hat\sigma}{\omega}\] \[= \epsilon_0 \left( 1 + i ^2 \frac{\omega_p ^2}{\omega ^2} \right)\] \[\epsilon = \epsilon_0 \left( 1 - \frac{\omega_p ^2}{\omega^2} \right)\] which gives susceptibility \[\hat \chi = - \frac{\omega_p ^2}{\omega ^2}\] which is what we found for a free oscillator.

To understand the wave dynamics, we need to look at the wave equation \[\pdv{^2 \vec E}{t^2} + c^2 \curl \curl \vec E + \omega_p ^2 \vec E = 0\] \[- \omega ^2 \hat \vec E - c^2 \vec k \cross \vec k \cross \hat \vec E + \omega_p ^2 \hat \vec E = 0\] \[\vec n \cross \vec n \cross \hat \vec E + (1 - \frac{\omega_p ^2}{\omega^2} ) \hat \vec E = 0\] where \[\vec n = \frac{c \vec k}{n}\] \[\vec n \cross \vec n \cross \hat \vec E + \frac{\epsilon}{\epsilon_0} \hat \vec E = 0\] In general, for non-isomorphic permittivity we’ll have a tensor result \[\vec n \cross \vec n \cross \hat \vec E + \frac{\hat{\overline \epsilon}}{\epsilon_0} \cdot \hat \vec E = 0\]

Using vector identity \[\vec n \cross \vec n \cross \vec E = \vec n (\vec n \cdot \vec E) - \vec E n ^2\] Converting to tensor notation, \[= n_i n_j E_j - E_i n^2\] \[= (n_i n_j - n ^2 I_{ij}) E_j\] and substituting back \[E_j \frac{\epsilon}{\epsilon_0} I_{ij} + \left( \frac{n_i n_j}{n^2} - I_{ij} \right) n^2 E_j = 0\] \[\left( \left( \frac{\epsilon(\omega)}{\epsilon_0} - n^2 \right) I_{ij} + n_i n_j \right) E_j = 0\] The big operator acting on \( \vec E \) we call the dispersion tensor. \[\overline D \cdot \vec E = 0\] For non-trivial solutions, we need the determinant \( \text{det}(\overline D) = 0 \) \[\hat{D}_{ij} = \left( \frac{\epsilon(\omega)}{\epsilon_0} - n^2 \right) I_{ij} + n_i n_j\]

As an example, if we have propagation in the \( x \)-direction \[\vec n = (n, 0, 0)\] Then the various components of \[(n_i n_j - n^2 I_{ij})\] are \[x \rightarrow \qquad n^2 - n^2 = 0\] \[y \rightarrow \qquad 0 - n^2 = -n^2\] \[z \rightarrow \qquad 0 - n^2 = - n^2\] so the dispersion tensor looks like \[\begin{bmatrix} \epsilon(\omega)/\epsilon_0 & 0 & 0 \\ 0 & \frac{\epsilon}{\epsilon_0} - n^2 & 0 \\ 0 & 0 & \frac{\epsilon}{\epsilon_0} \end{bmatrix}\] We don’t need to invert this, we just get three equations \[\hat E_x (\epsilon / \epsilon_0) = 0\] \[\hat E_y (\epsilon / \epsilon_0 - n^2) = 0\] \[E_z (\epsilon / \epsilon_0 - n^2) = 0\] In the parallel direction \( k_\parallel \), we get \( \epsilon = 0 \). In the transverse directions \( k_\perp \) we get \( \epsilon = n^2 \)

\[k_\parallel \quad 1 - \frac{\omega_p ^2}{\omega ^2} = 0\] \[k_{\perp, 1} \qquad 1 - \frac{\omega_p ^2}{\omega ^2} - \frac{c^2 k^2}{\omega^2} = 0\] \[k_{\perp, 2} \qquad 1 - \frac{\omega_p ^2}{\omega ^2} - \frac{c^2 k^2}{\omega^2} = 0\] We can solve these because they’re just functions of \( \omega^2 \). The first solution is \[\omega ^2 = \omega_p ^2\] which is a static plasma oscillation. In the other solutions, we get the ordinary plasma wave \[\omega ^2 = \omega _p ^2 + c^2 k^2\] This is what the dispersion relation becomes when light enters a plasma, from the free space relation \( \omega ^2 = c^2 k^2 \). The ordinary plasma wave is dispersive, since the phase and group velocities are not equal to each other.