2 Model

The TPD instability is a three-wave parametric instability in which an electromagnetic wave decays into a pair of plasma waves. Due to the Manley–Rowe relations, the frequencies and wavenumbers of the interacting waves must be matched (Michel Reference Michel2023). This localises the instability spatially near the quarter critical density – the point where the laser frequency is equal to twice the local plasma frequency (see figure 1a). Furthermore, since all of the waves are high in frequency, the ion motion can be neglected in the linear stage of the instability. The response of the electrons in the presence of the laser field can be analysed using non-relativistic fluid equations. Kinetic effects can be ignored provided $k \lambda _{De} \ll 1$. Relativistic effects are negligible under conventional hot-spot ignition ICF where the laser intensity is of order $I_{{L}} \sim 10^{15} \, \text {W}\,\text {cm}^{-2}$ and the laser wavelength is $\lambda _{{L}} \sim 350 \ {\mu }\text {m}$. This is because the quiver velocity of the electrons $\boldsymbol {v}_{\text {os}}(t,\boldsymbol {x}) = e \boldsymbol {A}_0/m_e$ in the presence of an electromagentic wave described by the vector potential $\boldsymbol {A}_0(t,\boldsymbol {x})$ is $v_{\text {os}}/c \sim 0.01$ under such conditions. Finally, for the motion of the ions to be neglected, we need the amplitude of electrostatic waves to be low enough such that $\sqrt {W_e} \ll k \lambda _{De}$, where $W_e = \tfrac 12 \epsilon _0 |\boldsymbol {\nabla } \varphi |^2/n_e T_e$. This ensures that the nonlinear terms which couple the electron and ion motion are of higher order and negligible. So, the following ordering will be assumed from now on:

(2.1)\begin{equation} \sqrt{\frac{\epsilon_0 |\boldsymbol{\nabla} \varphi|^2}{2n_e T_e}} \ll \frac{v_{\text{os}}}{c} \sim k \lambda_{De} \ll 1. \end{equation}

Assuming a purely electrostatic perturbation, linearising the continuity and momentum equations for the electron fluid gives (Liu & Rosenbluth Reference Liu and Rosenbluth1976; Simon et al. Reference Simon, Short, Williams and Dewandre1983)

(2.2)\begin{equation} \left.\begin{array}{@{}c@{}} \displaystyle \dfrac{\partial n}{\partial t} ={-} \nabla^2 \psi - \boldsymbol{v}_{\text{os}} \boldsymbol{{\cdot}} \boldsymbol{\nabla} n, \\ \displaystyle \dfrac{\partial \psi}{\partial t} = \omega_{\text{pe}}^{2} \nabla^{{-}2} n - 3 v_{\text{th}e}^{2} n - \boldsymbol{v}_{\text{os}} \boldsymbol{{\cdot}} \boldsymbol{\nabla} \psi. \end{array} \right\} \end{equation}

Here $n(t,\boldsymbol {x})$ is the electron density perturbation normalised to the equilibrium density $n_{0}$ which will be assumed uniform for simplicity, $\psi (t,\boldsymbol {x})$ is the velocity potential of the electrons meaning $\boldsymbol {u} = \boldsymbol {\nabla } \psi$, where $\boldsymbol {u}$ is the electron fluid velocity, $\omega _{\textrm {pe}}^{2} = e^2 n_0/\epsilon _0 m_e$ is the equilibrium plasma frequency and $v_{\textrm {th}e}^{2} = T_e/m_e$ is the electron thermal velocity. Since the goal is to develop a description of the instability under broad-bandwidth (stochastic) laser fields, $n, \psi$ and $\boldsymbol {v}_{\textrm {os}}$ are assumed to be random variables.

The wave equation governing the electron plasma waves is

(2.3)\begin{equation} \left( \frac{\partial ^2}{\partial t^2} + \omega_{\text{pe}}^{2} - 3 v_{\text{th}e}^2 \nabla^2 \right) n = \nabla^2 ( \boldsymbol{v}_{\text{os}} \boldsymbol{{\cdot}} \boldsymbol{\nabla} \psi) - \frac{\partial }{\partial t} ( \boldsymbol{v}_{\text{os}} \boldsymbol{{\cdot}} \boldsymbol{\nabla} n ), \end{equation}

where the right-hand side contains the TPD driving terms. One can express the driving terms through the Wigner functions describing the correlations between the pump field $\boldsymbol {v}_{\textrm {os}}$ and the plasma waves, as has been done in the case of inhomogeneous Navier–Stokes turbulence (Tsiolis, Zhou & Dodin Reference Tsiolis, Zhou and Dodin2020) (see Appendix B):

(2.4)\begin{equation} \boldsymbol{v}_{\text{os}} \boldsymbol{{\cdot}} \boldsymbol{\nabla} \psi = \textsf{i} \int \frac{\text{d} \boldsymbol{k}}{(2\pi)^3} \boldsymbol{k}^{\top} \star {\boldsymbol{\mathsf{W}}}_{\psi \boldsymbol{v}}^{\top}. \end{equation}

Here ${\boldsymbol{\mathsf{W}}}_{\psi \boldsymbol {v}}(\boldsymbol {x},\boldsymbol {k}) = \mathscr {W} [ \left | \psi \right \rangle \left \langle \boldsymbol {v}_{\textrm {os}} \right | ]$, with $\mathscr {W}$ being the Wigner transform, and the bra-ket notation is used as in Dodin (Reference Dodin2022); ${\boldsymbol{\mathsf{W}}}_{\psi \boldsymbol {v}}^{\intercal }$ denotes the transpose of ${\boldsymbol{\mathsf{W}}}_{\psi \boldsymbol {v}}$. The Wigner transform $\mathscr {W}[\hat {A}]$ of some generic operator $\hat {A}$ is called the Weyl symbol of $\hat {A}$, and is defined as

(2.5)\begin{equation} \textsf{A}(\boldsymbol{x},\boldsymbol{k}) \equiv \mathscr{W}[\hat{A}] \doteq \int \text{d} \boldsymbol{s} \, \text{e}^{- \textsf{i} \boldsymbol{k} \boldsymbol{{\cdot}} \boldsymbol{s}} \left\langle \boldsymbol{x} + \frac{\boldsymbol{s}}{2} \right| \hat{A} \left| \boldsymbol{x} - \frac{\boldsymbol{s}}{2} \right\rangle. \end{equation}

The Moyal star product $\star$ between two symbols is defined as $\textsf {A} \star \textsf {B} \doteq \mathscr {W}[\hat {A} \hat {B}]$; it is given by $\textsf {A}(\boldsymbol {x},\boldsymbol {k}) \star \textsf {B}(\boldsymbol {x},\boldsymbol {k}) = \textsf {A}(\boldsymbol {x},\boldsymbol {k})\, \textrm {e}^{\textsf {i} \hat {\mathcal {P}}/2} \textsf {B}(\boldsymbol {x},\boldsymbol {k})$, with $\hat {\mathcal {P}}$ being the Poisson bracket $\hat {\mathcal {P}} = \overset {{\scriptscriptstyle \leftarrow }}{\partial}_{\boldsymbol {x}} \boldsymbol {{\cdot }} \overset {{\scriptscriptstyle \rightarrow }}{\partial}_{\boldsymbol {k}} - \overset {{\scriptscriptstyle \leftarrow }}{\partial}_{\boldsymbol {k}} \boldsymbol {{\cdot }} \overset {{\scriptscriptstyle \rightarrow }}{\partial}_{\boldsymbol {x}}$, and the arrows indicating the directions in which the derivatives are acting. Element-wise application of the Moyal star product $\star$ and the usual matrix multiplication rules are assumed, meaning that for any two matrix symbols ${\boldsymbol{\mathsf{A}}}$ and ${\boldsymbol{\mathsf{B}}}$ one applies the Moyal product as follows: $( {\boldsymbol{\mathsf{A}}} \star {\boldsymbol{\mathsf{B}}} )_{ij} = \sum _k ({\boldsymbol{\mathsf{A}}})_{ik} \star ({\boldsymbol{\mathsf{B}}})_{kj}$. A brief introduction to the Weyl symbol calculus can be found in Appendix A. With these definitions, one can see that the quantity ${\boldsymbol{\mathsf{W}}}_{\psi \boldsymbol {v}}(\boldsymbol {x},\boldsymbol {k})$ is given by

(2.6)\begin{equation} {\boldsymbol{\mathsf{W}}}_{\psi \boldsymbol{v}}(\boldsymbol{x},\boldsymbol{k}) = \int \text{d} \boldsymbol{s} \, \text{e}^{- \textsf{i} \boldsymbol{k} \boldsymbol{{\cdot}} \boldsymbol{s}} \psi( \boldsymbol{x} + \boldsymbol{s}/2 ) \boldsymbol{v}_{\text{os}}^{{{\dagger}}} (\boldsymbol{x} - \boldsymbol{s}/2). \end{equation}

After ensemble averaging, ${\boldsymbol{\mathsf{W}}}_{\psi \boldsymbol {v}}(\boldsymbol {x},\boldsymbol {k})$ will represent the correlations of the laser pulse field $\boldsymbol {v}_{\textrm {os}}$ with the the velocity potential $\psi$. The second driving term is rewritten in a completely analogous manner leading to the quantity ${\boldsymbol{\mathsf{W}}}_{n \boldsymbol {v}}(\boldsymbol {x},\boldsymbol {k})$ which will represent the correlations between $\boldsymbol {v}_{\textrm {os}}$ and the density perturbation $n$.

Taking the Fourier transform of the wave equation, and ensemble averaging, leads to

(2.7)\begin{equation} D_e(\varOmega, \boldsymbol{K}) \mathring{\bar{n}} (\varOmega,\boldsymbol{K}) = \int \frac{\text{d} \boldsymbol{k}}{(2\pi)^3} \, \left( \boldsymbol{k} - \tfrac12 \boldsymbol{K} \right) \boldsymbol{{\cdot}} \left( \textsf{i} K^2 \mathring{\bar{{\boldsymbol{\mathsf{W}}}}}_{\psi \boldsymbol{v}} - \varOmega \mathring{\bar{{\boldsymbol{\mathsf{W}}}}}_{n \boldsymbol{v}} \right), \end{equation}

where $D_e(\varOmega, \boldsymbol {K}) = \varOmega ^2 - \omega _{\textrm {pe}}^{2} - 3 v_{\textrm {th}e}^{2} K^2 = \varOmega ^2 - \omega _{e\boldsymbol {K}}^2$ is the dispersion function for the electron plasma waves, and the Fourier transformed quantities are denoted as follows: $\mathring {n}(\varOmega,\boldsymbol {K}) = \int \textrm {d} t \textrm {d} \boldsymbol {x} \, n(t,\boldsymbol {x}) \,\textrm {e}^{\textsf {i} \varOmega t - \textsf {i} \boldsymbol {K} \boldsymbol {{\cdot }} \boldsymbol {x}}$. The overline represents ensemble averaging. Here one now faces the closure problem – to calculate the mean field $\bar {n}$, knowledge is needed of second-order quantities: $\bar {{\boldsymbol{\mathsf{W}}}}_{\psi \boldsymbol {v}}$ and $\bar {{\boldsymbol{\mathsf{W}}}}_{n \boldsymbol {v}}$. The powerful toolbox of the Weyl symbol calculus is now applied to derive and solve the equations governing these quantities.

The basic idea is as follows: if one writes down the governing equations of the system in Schrödinger form, then all of the second-order correlation functions describing the system will be governed by the Wigner–Moyal equation (see Appendix A). A similar procedure was carried out for the case of inhomogeneous fluid turbulence by Tsiolis et al. (Reference Tsiolis, Zhou and Dodin2020). The TPD equations (2.2) are first order in time and are readily put in Schrödinger form. The vector potential $\boldsymbol {A}_{0}$ describing the propagation of an electromagnetic mode in a plasma obeys a Klein–Gordon equation and therefore $( \partial _t^2 - c^2 \nabla ^2 + \omega _{\textrm {pe}}^{2} ) \boldsymbol {v}_{\textrm {os}} = 0$, since $\boldsymbol {v}_{\textrm {os}}$ differs from $\boldsymbol {A}_0$ only by a constant factor (Kruer Reference Kruer2019; Michel Reference Michel2023). There are no source or sink terms in this equation since the linear stage of the instability is considered and hence pump depletion is ignored. This second-order-in-time equation can be decomposed into two first-order ones by defining the auxiliary fields $\boldsymbol {\phi }, \boldsymbol {\chi } = \frac 12 ( \boldsymbol {v}_{\textrm {os}} \pm \textsf {i} \omega _{\textrm {pe}}^{-1} \partial _t \boldsymbol {v}_{\textrm {os}} )$ (Santos & Silva Reference Santos and Silva2005).

The system of the decomposed Klein–Gordon equation, together with the two TPD equations are then written as a Schrödinger equation, $\textsf {i} \partial _t \boldsymbol {\varPsi } = \widehat{\boldsymbol{\mathsf{H}}} \boldsymbol {\varPsi }$, with the following matrix Hamiltonian and state vector:

(2.8a)\begin{align} \widehat{\boldsymbol{\mathsf{H}}} = \begin{pmatrix} 0 & - \textsf{i} \nabla^2 & - \textsf{i} \boldsymbol{\eta}^{\top} & - \textsf{i} \boldsymbol{\eta}^{\top}\\ \textsf{i} ( \omega_{\text{pe}}^{2} \nabla^{{-}2} - 3 v_{\text{th}e}^{2} ) & 0 & - \textsf{i} \boldsymbol{u}^{\top} & - \textsf{i} \boldsymbol{u}^{\top} \\ 0 & 0 & - \dfrac{c^2}{2 \omega_{\text{pe}}} \nabla^2 + \omega_{\text{pe}} & - \dfrac{c^2}{2 \omega_{\text{pe}}} \nabla^2 \\ 0 & 0 & \dfrac{c^2}{2 \omega_{\text{pe}}} \nabla^2 & \dfrac{c^2}{2 \omega_{\text{pe}}} \nabla^2 - \omega_{\text{pe}} \end{pmatrix}, \quad \boldsymbol{\varPsi} = \begin{pmatrix} n \\ \psi \\ \boldsymbol{\phi} \\ \boldsymbol{\chi} \end{pmatrix}, \end{align}

where $\boldsymbol {\eta } \doteq \boldsymbol {\nabla } n$ and $\boldsymbol {u} \doteq \boldsymbol {\nabla } \psi$. The Wigner matrix ${\boldsymbol{\mathsf{W}}} = \mathscr {W} [ \left | \boldsymbol {\varPsi } \right \rangle \left \langle \boldsymbol {\varPsi } \right | ]$ satisfies the Wigner–Moyal equation:

(2.9)\begin{equation} \textsf{i} \partial_t {\boldsymbol{\mathsf{W}}} = {\boldsymbol{\mathsf{H}}} \star {\boldsymbol{\mathsf{W}}} - {\boldsymbol{\mathsf{W}}} \star {\boldsymbol{\mathsf{H}}}^{{{\dagger}}}, \end{equation}

with ${\boldsymbol{\mathsf{H}}} \doteq \mathscr {W}[\widehat{\boldsymbol{\mathsf{H}}}]$ being the symbol of the matrix Hamiltonian. It is useful to write out the contents of the Wigner matrix and the symbol of the matrix Hamiltonian explicitly:

(2.10)\begin{gather} {\boldsymbol{\mathsf{H}}} = \begin{pmatrix} 0 & \textsf{i} k^2 & - \textsf{i} \boldsymbol{\eta}^{\top} & - \textsf{i} \boldsymbol{\eta}^{\top}\\ - \textsf{i} k^{{-}2} \omega_{e\boldsymbol{k}}^2 & 0 & - \textsf{i} \boldsymbol{u}^{\top} & - \textsf{i} \boldsymbol{u}^{\top}\\ 0 & 0 & \dfrac{c^2}{2 \omega_{\text{pe}}} k^2 + \omega_{\text{pe}} & \dfrac{c^2}{2 \omega_{\text{pe}}} k^2 \\ 0 & 0 & - \dfrac{c^2}{2 \omega_{\text{pe}}} k^2 & - \dfrac{c^2}{2 \omega_{\text{pe}}} k^2 - \omega_{\text{pe}} \end{pmatrix}, \end{gather}

(2.11)\begin{gather}{\boldsymbol{\mathsf{W}}} = \begin{pmatrix} \textsf{W}_{nn} & \textsf{W}_{n \psi} & {\boldsymbol{\mathsf{W}}}_{n \boldsymbol{\phi}} & {\boldsymbol{\mathsf{W}}}_{n \boldsymbol{\chi}} \\ \textsf{W}_{\psi n} & \textsf{W}_{\psi \psi } & {\boldsymbol{\mathsf{W}}}_{\psi \boldsymbol{\phi}} & {\boldsymbol{\mathsf{W}}}_{\psi \boldsymbol{\chi}} \\ {\boldsymbol{\mathsf{W}}}_{\boldsymbol{\phi} n} & {\boldsymbol{\mathsf{W}}}_{\boldsymbol{\phi} \psi} & {\boldsymbol{\mathsf{W}}}_{\boldsymbol{\phi} \boldsymbol{\phi}} & {\boldsymbol{\mathsf{W}}}_{\boldsymbol{\phi} \boldsymbol{\chi}} \\ {\boldsymbol{\mathsf{W}}}_{\boldsymbol{\chi} n} & {\boldsymbol{\mathsf{W}}}_{\boldsymbol{\chi} \psi} & {\boldsymbol{\mathsf{W}}}_{\boldsymbol{\chi} \boldsymbol{\phi}} & {\boldsymbol{\mathsf{W}}}_{\boldsymbol{\chi} \boldsymbol{\chi}} \end{pmatrix}, \end{gather}

where the quantities which have a vector index in the second slot such as ${\boldsymbol{\mathsf{W}}}_{n \boldsymbol {\phi }}$ are row vectors; similarly ${\boldsymbol{\mathsf{W}}}_{\boldsymbol {\phi }n}$ is a column vector; and quantities such as ${\boldsymbol{\mathsf{W}}}_{\boldsymbol {\phi } \boldsymbol {\phi }}$ with two vector indices are matrices.

Note that the relevant Wigner functions in (2.7) are related to the components of this Wigner matrix above through ${\boldsymbol{\mathsf{W}}}_{n \boldsymbol {v}} = {\boldsymbol{\mathsf{W}}}_{n \boldsymbol {\phi }} + {\boldsymbol{\mathsf{W}}}_{n \boldsymbol {\chi }}$ and ${\boldsymbol{\mathsf{W}}}_{\psi \boldsymbol {v}} = {\boldsymbol{\mathsf{W}}}_{\psi \boldsymbol {\phi }} + {\boldsymbol{\mathsf{W}}}_{\psi \boldsymbol {\chi }}$, and so one needs only to compute the evolution equations of the upper right $2\times 2$ corner of ${\boldsymbol{\mathsf{W}}}$. Doing so and taking sums and differences of the equations in the resulting system, one gets equations for ${\boldsymbol{\mathsf{W}}}_{n \boldsymbol {v}}$ and ${\boldsymbol{\mathsf{W}}}_{\psi \boldsymbol {v}}$, which upon ensemble averaging results in the following system:

(2.12)\begin{equation} \left.\begin{array}{@{}c@{}} \textsf{i} \partial_t \bar{{\boldsymbol{\mathsf{W}}}}_{n \boldsymbol{v}} = \textsf{i} k^2 \star \bar{{\boldsymbol{\mathsf{W}}}}_{\psi \boldsymbol{v}} - \textsf{i} \bar{\boldsymbol{\eta}}^{\top} \star \bar{{\boldsymbol{\mathsf{W}}}}_{\boldsymbol{v} \boldsymbol{v}} - \omega_{\text{pe}} \bar{{\boldsymbol{\mathsf{B}}}}_{n \boldsymbol{v}}, \\ \textsf{i} \partial_t \bar{{\boldsymbol{\mathsf{B}}}}_{n \boldsymbol{v}} = \textsf{i} k^2 \star \bar{{\boldsymbol{\mathsf{B}}}}_{\psi \boldsymbol{v}} - \textsf{i} \bar{\boldsymbol{\eta}}^{\top} \star \bar{{\boldsymbol{\mathsf{B}}}}_{\boldsymbol{v} \boldsymbol{v}} - \omega_{\text{pe}}^{{-}1} \bar{{\boldsymbol{\mathsf{W}}}}_{n \boldsymbol{v}} \star \omega_{\boldsymbol{k}}^2, \\ \textsf{i} \partial_t \bar{{\boldsymbol{\mathsf{W}}}}_{\psi \boldsymbol{v}} ={-} \textsf{i} k^{{-}2} \omega_{e\boldsymbol{k}}^2 \star \bar{{\boldsymbol{\mathsf{W}}}}_{n \boldsymbol{v}} - \textsf{i} \bar{\boldsymbol{u}}^{\top} \star \bar{{\boldsymbol{\mathsf{W}}}}_{\boldsymbol{v} \boldsymbol{v}} - \omega_{\text{pe}} \bar{{\boldsymbol{\mathsf{B}}}}_{\psi \boldsymbol{v}}, \\ \textsf{i} \partial_t \bar{{\boldsymbol{\mathsf{B}}}}_{\psi \boldsymbol{v}} ={-} \textsf{i} k^{{-}2} \omega_{e\boldsymbol{k}}^2 \star \bar{{\boldsymbol{\mathsf{B}}}}_{n \boldsymbol{v}} - \textsf{i} \bar{\boldsymbol{u}}^{\top} \star \bar{{\boldsymbol{\mathsf{B}}}}_{\boldsymbol{v} \boldsymbol{v}} - \omega_{\text{pe}}^{{-}1} \bar{{\boldsymbol{\mathsf{W}}}}_{\psi \boldsymbol{v}} \star \omega_{\boldsymbol{k}}^2. \end{array}\right\} \end{equation}

Here $\omega _{\boldsymbol {k}}^2 = \omega _{\textrm {pe}}^{2} + k^2 c^2$ is the electromagnetic wave frequency. All of the ${\boldsymbol{\mathsf{B}}}$ quantities are not of interest and simply facilitate the calculation; they are defined as follows: ${\boldsymbol{\mathsf{B}}}_{n \boldsymbol {v}} \doteq {\boldsymbol{\mathsf{W}}}_{n \boldsymbol {\phi }} - {\boldsymbol{\mathsf{W}}}_{n \boldsymbol {\chi }}, {\boldsymbol{\mathsf{B}}}_{\psi \boldsymbol {v}} \doteq {\boldsymbol{\mathsf{W}}}_{\psi \boldsymbol {\phi }} - {\boldsymbol{\mathsf{W}}}_{\psi \boldsymbol {\chi }}, {\boldsymbol{\mathsf{B}}}_{\boldsymbol {v} \boldsymbol {v}} \doteq {\boldsymbol{\mathsf{W}}}_{\boldsymbol {\phi } \boldsymbol {\phi }} + {\boldsymbol{\mathsf{W}}}_{\boldsymbol {\chi } \boldsymbol {\phi }} - {\boldsymbol{\mathsf{W}}}_{\boldsymbol {\phi } \boldsymbol {\chi }} - {\boldsymbol{\mathsf{W}}}_{\boldsymbol {\chi } \boldsymbol {\chi }}$. Here, the fact that ${\boldsymbol{\mathsf{W}}}_{\boldsymbol {v} \boldsymbol {v}} = {\boldsymbol{\mathsf{W}}}_{\boldsymbol {\phi } \boldsymbol {\phi }} + {\boldsymbol{\mathsf{W}}}_{\boldsymbol {\chi } \boldsymbol {\phi }} + {\boldsymbol{\mathsf{W}}}_{\boldsymbol {\phi } \boldsymbol {\chi }} + {\boldsymbol{\mathsf{W}}}_{\boldsymbol {\chi } \boldsymbol {\chi }}$ is used, which follows from the linearity of the Wigner transform.

The crucial step, which is the essence of the CE2 closure, is the splitting of the average of the product, into the product of the averages, for the third-order quantities: $\overline {\boldsymbol {\eta }^{\top } \star {\boldsymbol{\mathsf{W}}}} = \bar {\boldsymbol {\eta }}^{\top } \star \bar {{\boldsymbol{\mathsf{W}}}}$. This effectively assumes the third-order cumulant $\overline { \tilde {\boldsymbol {\eta }}^{\top } \star \widetilde {{\boldsymbol{\mathsf{W}}}}}$ to be negligible (the tilde denotes the fluctuating part of the quantity) (Krommes & Parker Reference Krommes and Parker2019). It is exactly zero for fields obeying jointly normal (Gaussian multivariate) statistics. In contrast to turbulence, here this assumption is easier to justify. One does not expect the interactions between the fluctuations to be of interest, which is what is being neglected when one discards higher-order cumulants and assumes Gaussianity. They would not be of interest since such effects are what leads to fluctuation intermittency, or are responsible for the typical turbulent energy cascades, neither of which is assumed here to be of relevance in the linear stage of the instability. The fluctuations in the electric field of the laser pulse are prescribed in a sense (their power spectrum $\bar {{\boldsymbol{\mathsf{W}}}}_{\boldsymbol {v} \boldsymbol {v}}$ is given), and are expected to be normal for both temporally incoherent pulses (Goodman Reference Goodman2015), as well as transversely incoherent fields produced by a random phase plate for example (Rose & DuBois Reference Rose and DuBois1993; Garnier Reference Garnier1999), where in both cases this is a consequence of the central limit theorem. Hence the driving field $\boldsymbol {v}_{\textrm {os}}$ is safely assumed a Gaussian random process. It should be noted that the Wigner–Moyal formalism is also very useful in deriving more sophisticated closures such as the quasi-normal one as shown by Ruiz et al. (Reference Ruiz, Glinsky and Dodin2019).

The Wigner function is bilinear and therefore ${\boldsymbol{\mathsf{B}}}_{\boldsymbol {v} \boldsymbol {v}} = {\boldsymbol{\mathsf{W}}}_{(\boldsymbol {\phi } + \boldsymbol {\chi )}(\boldsymbol {\phi } - \boldsymbol {\chi )}}$. If the laser field $\boldsymbol {v}_{\textrm {os}}$ is statistically stationary, it is shown in Appendix C that the ensemble-averaged Wigner function $\bar {{\boldsymbol{\mathsf{W}}}}_{\boldsymbol {v}\boldsymbol {v}}$ and the associated quantity $\bar {{\boldsymbol{\mathsf{B}}}}_{\boldsymbol {v} \boldsymbol {v}}$ are related as follows: $\bar {{\boldsymbol{\mathsf{B}}}}_{\boldsymbol {v} \boldsymbol {v}}(\boldsymbol {k}) = \omega _{\textrm {pe}}^{-1} \omega _{\boldsymbol {k}} \bar {{\boldsymbol{\mathsf{W}}}}_{\boldsymbol {v}\boldsymbol {v}}(\boldsymbol {k})$. This is an important step in the derivation as it makes all symbol products in (2.12) have one of the quantities being a function of $\boldsymbol {x}$ or $\boldsymbol {k}$ only. Taking the Fourier transform of the system results in

(2.13)\begin{equation} \left.\begin{array}{@{}c@{}} -\varOmega \mathring{\bar{{\boldsymbol{\mathsf{W}}}}}_{n \boldsymbol{v}} = \textsf{i} k^2_- \mathring{\bar{{\boldsymbol{\mathsf{W}}}}}_{\psi \boldsymbol{v}} - \textsf{i} \mathring{\bar{\boldsymbol{\eta}}}^{\top} \bar{{\boldsymbol{\mathsf{W}}}}_{\boldsymbol{v} \boldsymbol{v}}^{+} - \omega_{\text{pe}} \mathring{\bar{{\boldsymbol{\mathsf{B}}}}}_{n \boldsymbol{v}}, \\ -\varOmega \mathring{\bar{{\boldsymbol{\mathsf{B}}}}}_{n \boldsymbol{v}} = \textsf{i} k_-^2 \mathring{\bar{{\boldsymbol{\mathsf{B}}}}}_{\psi \boldsymbol{v}} - \textsf{i} \omega_{\text{pe}}^{{-}1} \omega_{\boldsymbol{k}_+} \mathring{\bar{\boldsymbol{\eta}}}^{\top} \bar{{\boldsymbol{\mathsf{W}}}}_{\boldsymbol{v} \boldsymbol{v}}^{+} - \omega_{\text{pe}}^{{-}1} \omega_{\boldsymbol{k}_+}^2 \mathring{\bar{{\boldsymbol{\mathsf{W}}}}}_{n \boldsymbol{v}}, \\ -\varOmega \mathring{\bar{{\boldsymbol{\mathsf{W}}}}}_{\psi \boldsymbol{v}} ={-} \textsf{i} k^{{-}2}_- \omega_{e\boldsymbol{k}_-}^2 \mathring{\bar{{\boldsymbol{\mathsf{W}}}}}_{n \boldsymbol{v}} - \textsf{i} \mathring{\bar{\boldsymbol{u}}}^{\top} \bar{{\boldsymbol{\mathsf{W}}}}_{\boldsymbol{v} \boldsymbol{v}}^{+} - \omega_{\text{pe}} \mathring{\bar{{\boldsymbol{\mathsf{B}}}}}_{\psi \boldsymbol{v}}, \\ -\varOmega \mathring{\bar{{\boldsymbol{\mathsf{B}}}}}_{\psi \boldsymbol{v}} ={-} \textsf{i} k^{{-}2}_- \omega_{e\boldsymbol{k}_-}^2 \mathring{\bar{{\boldsymbol{\mathsf{B}}}}}_{n \boldsymbol{v}} - \textsf{i} \omega_{\text{pe}}^{{-}1} \omega_{\boldsymbol{k}_+} \mathring{\bar{\boldsymbol{u}}}^{\top} \bar{{\boldsymbol{\mathsf{W}}}}_{\boldsymbol{v} \boldsymbol{v}}^{+} -\omega_{\text{pe}}^{{-}1} \omega_{\boldsymbol{k}_+}^2 \mathring{\bar{{\boldsymbol{\mathsf{W}}}}}_{\psi \boldsymbol{v}}, \end{array}\right\} \end{equation}

where the quantities with superscripts $\pm$ are evaluated at $\boldsymbol {k}_{\pm } = \boldsymbol {k} \pm \tfrac 12 \boldsymbol {K}, \mathring {\bar {\boldsymbol {\eta }}}$ and $\mathring {\bar {\boldsymbol {u}}}$ are functions of $\boldsymbol {K}$ only (the Fourier image of $\boldsymbol {x}$) and all other quantities are functions of both $\boldsymbol {k}$ and $\boldsymbol {K}$.

In order to keep the model self-consistent under the assumed ordering, the growth rate of the instability can be calculated only up to order $v_{\textrm {os}}/c$ and higher-order terms need to be discarded (Kruer Reference Kruer2019). The way this works in the present case is as follows. To close the system (2.13), $\mathring {\boldsymbol {u}}$ can be expressed in terms of $\mathring {\boldsymbol {\eta }}$ and $\mathring {{\boldsymbol{\mathsf{W}}}}_{\psi \boldsymbol {v}}$ through the second equation in the TPD system (2.2). Then the equations containing $\mathring {\boldsymbol {u}}$ will involve terms which are $\propto \mathring {{\boldsymbol{\mathsf{W}}}}_{\psi \boldsymbol {v}} \mathring {{\boldsymbol{\mathsf{W}}}}_{\boldsymbol {v} \boldsymbol {v}}$. As in the standard derivation in Kruer (Reference Kruer2019), this introduces terms of order $(v_{\textrm {os}}/c)^2 \ll 1$ in the growth rate which are ignored, and therefore in (2.13) one uses the approximation $\mathring {\boldsymbol {u}} = \textsf {i} (\varOmega /K^2) \mathring {\boldsymbol {\eta }}$.

Solving the system for the quantities of interest, $\mathring {\bar {{\boldsymbol{\mathsf{W}}}}}_{\psi \boldsymbol {v}}$ and $\mathring {\bar {{\boldsymbol{\mathsf{W}}}}}_{n \boldsymbol {v}}$, in terms of the laser field $\bar {{\boldsymbol{\mathsf{W}}}}_{\boldsymbol {v} \boldsymbol {v}}$ and substituting them into (2.7) results in the homogeneous broadband TPD dispersion relation:

(2.14)\begin{equation} D_e(\varOmega,\boldsymbol{K}) = \int \frac{\text{d} \boldsymbol{k}}{(2\pi)^3} \frac{( k_-^2 \varOmega - K^2 \varOmega_- )^2 }{K^2 k_-^2 D_e(\varOmega_-,\boldsymbol{k}_-)} \boldsymbol{k}_- \boldsymbol{K}^{\top} \boldsymbol{:} \bar{{\boldsymbol{\mathsf{W}}}}_{\boldsymbol{v} \boldsymbol{v}} (\boldsymbol{k}), \end{equation}

where $\varOmega _- = \omega _{\boldsymbol {k}} - \varOmega$, the integral is to be taken along the Landau contour, $\boldsymbol {k}_- \doteq \boldsymbol {k} - \boldsymbol {K}$ has been redefined and the colon denotes double matrix contraction.Footnote ³ This is the main result of the new model. For a statistically stationary random process, the two-point correlation function and the power spectrum form a Fourier pair (Wiener–Khinchin theorem) (Goodman Reference Goodman2015). So one can notice that the expression has a familiar form and a simple physical interpretation: the broadband dispersion relation is the monochromatic one integrated over the laser field power spectrum $\bar {{\boldsymbol{\mathsf{W}}}}_{\boldsymbol {v} \boldsymbol {v}} (\boldsymbol {k})$, given by

(2.15)\begin{equation} \bar{{\boldsymbol{\mathsf{W}}}}_{\boldsymbol{v} \boldsymbol{v}} (\boldsymbol{k}) = \int \text{d} \boldsymbol{s} \, \text{e}^{- \textsf{i} \boldsymbol{k} \boldsymbol{{\cdot}} \boldsymbol{s}} \overline{\boldsymbol{v}_{\text{os}}(\boldsymbol{x}+\boldsymbol{s}/2) \boldsymbol{v}^{\dagger}_{\text{os}}(\boldsymbol{x}-\boldsymbol{s}/2)}. \end{equation}

This dispersion relation allows one to study the homogeneous TPD under laser fields with arbitrary power spectra.

3 Applications

3.1 Monochromatic beams

The broadband TPD dispersion relation (2.14) reduces to the well-known result for a monochromatic plane wave. Consider a pump of the form $\boldsymbol {v}_{\textrm {os}}(t,\boldsymbol {x}) = \tfrac 12 v_{\textrm {os}} \boldsymbol {\sigma } \,\textrm {e}^{\textsf {i} \boldsymbol {k}_0 \boldsymbol {{\cdot }} \boldsymbol {x} - \textsf {i} \omega _0t} + \textrm {c.c.}$, where $\omega _0 \equiv \omega _{\boldsymbol {k}_0}$ is the laser frequency and $\boldsymbol {\sigma }$ is the laser polarisation. The polarisation $\boldsymbol {\sigma }$ is allowed to be complex to take into account circularly polarised laser beams as well. The Wigner function of this field isFootnote ⁴

(3.1)\begin{equation} \bar{{\boldsymbol{\mathsf{W}}}}_{\boldsymbol{vv}}(\boldsymbol{k}) = \tfrac14 (2\pi)^3 v_{\text{os}}^2 [ \boldsymbol{\sigma} \boldsymbol{\sigma}^{{{\dagger}}} \delta(\boldsymbol{k} - \boldsymbol{k}_0) + \boldsymbol{\sigma}^* \boldsymbol{\sigma}^{\top} \delta(\boldsymbol{k} + \boldsymbol{k}_0)]. \end{equation}

Substituting this in (2.14), and as usual ignoring the anti-Stokes term for being off-resonant, reduces to the familiar expression (see Kruer Reference Kruer2019):

(3.2)\begin{equation} D_e(\varOmega,\boldsymbol{K}) D_e(\omega_0 - \varOmega,\boldsymbol{K}-\boldsymbol{k}_0) = \frac{\omega_{\text{pe}}^2 v_{\text{os}}^2 |\boldsymbol{K} \boldsymbol{\cdot \boldsymbol{\sigma}}|^2}{4} \left( \frac{K^2 - |\boldsymbol{K}-\boldsymbol{k}_0|^2}{K |\boldsymbol{K}-\boldsymbol{k}_0|} \right)^{2}, \end{equation}

where one has used the approximation $\varOmega \simeq \varOmega _{-} \simeq \omega _{\textrm {pe}}$ in the numerator.

In the case of multiple monochromatic non-interfering laser beams, the Wigner function is (ignoring the anti-Stokes terms)

(3.3)\begin{equation} \bar{{\boldsymbol{\mathsf{W}}}}_{\boldsymbol{vv}}(\boldsymbol{k}) = \sum_{i} \tfrac14 (2\pi)^3 v_{\text{os},i}^2 \boldsymbol{\sigma}_i \boldsymbol{\sigma}_i^{{{\dagger}}} \delta(\boldsymbol{k} - \boldsymbol{k}_{0i}). \end{equation}

The dispersion relation in this case becomes

(3.4)\begin{equation} D_e(\varOmega,\boldsymbol{K}) = \sum_{i} \left| \frac{|\boldsymbol{K}-\boldsymbol{k}_{0i}|^2 - K^2}{2 K |\boldsymbol{K}-\boldsymbol{k}_{0i}|} (\boldsymbol{K} \boldsymbol{{\cdot}} \boldsymbol{\sigma}_{i}) v_{\text{os},i} \right|^2 \frac{ \omega_{\text{pe}}^2}{D_e(\omega_0-\varOmega,\boldsymbol{K}-\boldsymbol{k}_{0i})}. \end{equation}

This agrees with the result of Michel et al. (Reference Michel, Maximov, Short, Delettrez, Edgell, Hu, Igumenshchev, Myatt, Solodov, Stoeckl, Yaakobi and Froula2013).

3.2 Single temporally incoherent beam

Next, consider a single temporally incoherent laser beam. The Wigner function describing such a field is of the form (again ignoring the anti-Stokes term)

(3.5)\begin{equation} \bar{{\boldsymbol{\mathsf{W}}}}_{\boldsymbol{vv}}(\boldsymbol{k}) = \tfrac14 (2\pi)^3 v_{\text{os}}^2 \boldsymbol{\sigma} \boldsymbol{\sigma}^{{{\dagger}}} G ( k_\parallel ) \delta(\boldsymbol{k}_{{\perp}}), \end{equation}

where $G(k_{\parallel })$ is the power spectrum centred around $k_0$ and normalised such that it integrates to unity. Also, $k_\parallel \doteq \boldsymbol {k} \boldsymbol {{\cdot }} \hat {\boldsymbol {\kappa }}, \boldsymbol {k}_\perp \doteq \boldsymbol {k} - k_\parallel \hat {\boldsymbol {\kappa }}$ with $\hat {\boldsymbol {\kappa }} \doteq \boldsymbol {k}_0/k_0$ being the unit vector in the direction of propagation. For $\Delta \omega / \omega _0 \ll 1$, the wavenumber spread $\Delta k$ is related to the frequency bandwidth through $\Delta \omega = v_{g0} \Delta k$, where $v_{g0} = \partial \omega _{\boldsymbol {k}} / \partial k |_{k_{0}}$ is the group velocity of electromagnetic waves at the peak of the power spectrum. Most commonly one considers broadband lasers with top-hat (flat), Lorentzian or Gaussian power spectra (Follett et al. Reference Follett, Shaw, Myatt, Dorrer, Froula and Palastro2019). Upon substitution of (3.5) in (2.14), the dispersion relation has the form

(3.6)\begin{equation} D_e(\varOmega,\boldsymbol{K}) = \frac{K_{{\perp}}^2v_{\text{os}}^2}{4} \mathscr{I}. \end{equation}

Here, in the same way as before $\boldsymbol {K}_{\perp } \doteq \boldsymbol {K} - \boldsymbol {K}\boldsymbol {{\cdot }} \hat {\boldsymbol {\kappa }}$. The integral $\mathscr {I}$ is given by

(3.7)\begin{equation} \mathscr{I} = \int_{C_{L}} \frac{\text{d} \omega_{k_{{\parallel}}}}{v_{g}(\omega_{k_{{\parallel}}})} \frac{ \textsf{C} (\omega_{k_{{\parallel}}} ) G (\omega_{k_{{\parallel}}} ) }{ ( \omega_{k_{{\parallel}}} - \varOmega )^2 - \omega_{e\boldsymbol{k}_-}^2 }, \end{equation}

in which the integration variables have been changed from $\boldsymbol {k}$ to $\omega _{\boldsymbol {k}}$ in order to deal with the pole in the integrand more easily; note that the integral should be taken along the Landau contour $C_{{L}}$. The function $\textsf {C}$ has been defined as $\textsf {C} (k_{\parallel }) \doteq (K^2 \varOmega _- - k_-^2 \varOmega )^2/(K k_-)^2$. Note that $\textsf {C}$ is a function of $\varOmega$ and $\boldsymbol {K}$ as well, but those have been dropped for brevity. Of course here by $k_-$ one means $k_- = |\boldsymbol {K} - k_{\parallel } \hat {\boldsymbol {\kappa }}|$. In order not to make the notation too cumbersome, functions such as $\textsf {C}$, the spectrum $G$, or $\omega _{e\boldsymbol {k}_-}$ which are defined as functions of $k_{\parallel }$, one evaluates in terms of the frequency implicitly as follows: $\textsf {C}(\omega _{k_\parallel }) \equiv \textsf {C}[\textsf {K}(\omega _{k_\parallel })]$, where $\textsf {K}(\omega _{\boldsymbol {k}}) \doteq c^{-1} \sqrt {\omega _{\boldsymbol {k}}^2 - \omega _{\textrm {pe}}^2} = k$ allows one to go from the laser frequency to the laser wavenumber.

In the monochromatic case, the TPD growth rate maximises to a value of $\gamma _0 = \tfrac 14 k_0 v_{\textrm {os}}$. This tells one that $\gamma _0/\omega _{\textrm {pe}}\ll 1$ since $\gamma _0/\omega _{\textrm {pe}} \sim k_0 v_{\textrm {os}}/\omega _{\textrm {pe}} \sim v_{\textrm {os}}/c \ll 1$. The right-hand side of (3.6) is of order $\gamma _0^2 \ll \omega _{\textrm {pe}}^2$. Therefore it is possible to assume that the real part of the frequency satisfies the linear dispersion relation: $\varOmega = \omega _{e\boldsymbol {K}} + \textsf {i} \gamma$. With this the growth rate becomes

(3.8)\begin{equation} \gamma = \frac{K_{{\perp}}^2v_{\text{os}}^2}{4} \frac{1}{2\omega_{e\boldsymbol{K}}} \text{Im}[\mathscr{I}], \end{equation}

and the integral $\mathscr {I}$ is

(3.9)\begin{equation} \mathscr{I} = \int_{C_{L}} \frac{\text{d} \omega_{k_{{\parallel}}}}{v_{g}(\omega_{k_{{\parallel}}})} \frac{ \textsf{C} (\omega_{k_{{\parallel}}} ) G (\omega_{k_{{\parallel}}} ) }{ ( \omega_{k_{{\parallel}}} - \omega_{e\boldsymbol{K}} - \textsf{i} \gamma - \omega_{e\boldsymbol{k}_-} ) ( \omega_{k_{{\parallel}}} - \omega_{e\boldsymbol{K}} - \textsf{i} \gamma + \omega_{e\boldsymbol{k}_-} )}. \end{equation}

The integrand of $\mathscr {I}$ has one pole of interest which occurs at a wavenumber $k_{{p}}(\boldsymbol {K})$ such that $\omega _{k_{{p}}} - \omega _{e\boldsymbol {K}} - \omega _{e\boldsymbol {k}_{-}} = 0$. This is the frequency-matching condition which for a given $k_{{p}}$ defines a circle in $\boldsymbol {K}$-space centred around $(K_{\parallel },K_{\perp }) = (k_0/2,0)$. In the limit of large bandwidth $\Delta \omega \gg \gamma$ one evaluates $\mathscr {I}$ using Plemelj's formula:

(3.10) \begin{equation} \lim_{\varepsilon \to 0^+} \int_{-\infty}^{\infty} \frac{f(x)}{x-\zeta \mp \textsf{i} \varepsilon} \, \text{d}\kern0.07em x = \mathscr{P} \int_{-\infty}^{\infty} \frac{f(x)}{x-\zeta} \, \text{d}\kern0.07em x \pm \textsf{i} \pi f(\zeta), \end{equation}

with $\mathscr {P} \displaystyle\int$ denoting the principal value integral and $\zeta, \varepsilon \in \mathbb{\text{R}}$. Physically this is due to the fact that for small growth rates, the resonance is very sharp and the instability is driven only by the part of the power spectrum which can perfectly satisfy the matching conditions. Evaluating the imaginary part at the pole:

(3.11)\begin{equation} \text{Im}[\mathscr{I}] = \pi \frac{\textsf{C}(k_{{p}}) G(k_{{p}}) }{ v_{g}(k_{{p}}) ( \omega_{k_{{p}}} - \omega_{e\boldsymbol{K}} + \omega_{e\boldsymbol{k}_-} )} \simeq \pi \frac{\textsf{C}(k_{{p}}) G(k_{{p}}) }{ v_{g0} \,\omega_{0} }. \end{equation}

This gives the growth rate of the TPD instability in a uniform plasma in the case of a single temporally incoherent laser beam:

(3.12)\begin{equation} \gamma_{\boldsymbol{K}} = \frac{\pi G(k_{{p}})}{v_{g0}} \gamma^2_{0\boldsymbol{K}}, \end{equation}

where one uses the fact that $({K_{\perp }^2v_{\textrm {os}}^2}/{8 \omega _{e\boldsymbol {K}} \omega _0}) \textsf {C}(k_{{p}},\boldsymbol {K}) \simeq ({K_{\perp }^2v_{\textrm {os}}^2}/{16})({(K^2 - |\boldsymbol {K} - k_{{p}} \hat {\boldsymbol {\kappa }}|^2)^2}/{K^2 |\boldsymbol {K} - k_{{p}} \hat {\boldsymbol {\kappa }}|^2}) = \gamma _{0\boldsymbol {K}}^2$, and $\gamma _{0\boldsymbol {K}}$ is the monochromatic growth rate. The reader is reminded here that the wavenumber at which the resonance occurs $k_{{p}}$ is a function of $\boldsymbol {K}$ and is determined from the frequency-matching condition: $\omega _{k_{{p}}} - \omega _{e\boldsymbol {K}} - \omega _{e\boldsymbol {k}_{-}} = 0$.

Consider next the maximum value that $\gamma _{\boldsymbol {K}}$ attains over all wavenumbers $\boldsymbol {K}$. Evaluating that for the top-hat and Lorentzian power spectra: for the top hat $G(k_0) = 1/\Delta k$ and so the growth rate is $\gamma _{\textrm {TH}} = \pi \gamma _0^2 / \Delta \omega$. For the Lorentzian $G(k) = (1/\pi ) ({\Delta k/2}/({(k-k_0)^2 + (\Delta k /2)^2}))$ one has $G(k_0) = 2/\pi \Delta k$ and so $\gamma _{{L}} = 2 \gamma _0^2/\Delta \omega$. Here $\gamma _0 = \tfrac 14 k_0 v_{\textrm {os}}$ is the value at which $\gamma _{0\boldsymbol {K}}$ maximises. These results agree with the well-known scaling of the growth rate as $\gamma \sim \gamma _0^2/\Delta \omega$, but also include the dependence of the result on the laser spectral shape. It is interesting to note that while the coherence times for the two power spectra are (for the top hat and Lorentzian, respectively) $\tau _c = 2\pi /\Delta \omega, 2/\Delta \omega$ (see also § 3.3 for a definition of $\tau _{{c}}$) (Goodman Reference Goodman2015; Follett et al. Reference Follett, Shaw, Myatt, Dorrer, Froula and Palastro2019), the growth rates differ by less than a factor of $\pi$. This shows that at least in the homogeneous case, it is not the coherence time per se which is the most relevant metric, but rather the fraction of laser energy which satisfies the resonance conditions. At first glance this is somewhat at odds with the results of Follett et al. (Reference Follett, Shaw, Myatt, Dorrer, Froula and Palastro2019), which found that the coherence time provides a universal scaling for the absolute Raman and TPD thresholds, meaning that the thresholds were the same for fields with the same coherence times but different spectral shapes. More formally, it appears that the peak growth rates of the instability scale with $G(k_0)$, and the absolute thresholds with $\int |G(k)|^2 \,\textrm {d} k$. The reason for this difference in behaviour is unknown but the problem of an absolute instability near the quarter critical density in an inhomogeneous plasma is significantly more complicated and there is little reason to expect that it should be affected by bandwidth in the same way as the temporal growth rate in a homogeneous plasma, which has been calculated here.

Having explored how bandwidth affects the maximum value the growth rate attains, one now turns to how it modifies the spectrum of excited plasma waves. First in the monochromatic case, plasma waves are driven at wavenumbers near the intersection of the circles defined by the frequency-matching condition, and the hyperbolas defined by $K_{\perp }^2 = K_{\parallel }(K_{\parallel } - k_0)$ at which the monochromatic growth rate $\gamma _{0\boldsymbol {K}}$ maximises. This is shown in the left-hand panel in figure 2. In the middle and right-hand panels the growth rate $\gamma _{\boldsymbol {K}}$ is presented, for a Lorentzian power spectrum of varying bandwidths. One sees that increasing the laser bandwidth significantly broadens the range of unstable wavenumbers excited by the TPD instability. In the right-hand panel virtually the entire TPD hyperbolas are observed since the ultra-large laser bandwidth allows all relevant wavenumbers to satisfy the frequency-matching conditions. What is most interesting is that even for relatively modest amounts of bandwidth such as $20 \ \textrm {THz}$ (middle panel) the broadening of the spectrum is substantial. Such effects may change the nature of the instability. For example, in an inhomogeneous plasma, at lower densities the TPD modes are fairly high in $\boldsymbol {K}$ as illustrated by the left-hand panel in figure 2, and tend to be convective. Absolute modes occur near $|\boldsymbol {K}| \to 0$, which would typically be excited only in regions very close to the quarter critical density (Yan, Maximov & Ren Reference Yan, Maximov and Ren2010). One sees that laser bandwidth favours the absolute instability at lower densities too.

Figure 2. Effect of laser bandwidth on the range of unstable wavenumbers excited by the TPD instability as driven by a single broadband laser beam. The laser has a Lorentzian power spectrum, an intensity of $I_{{L}} = 10^{15} \, \textrm {W}\, \textrm {cm}^{-2}$ and wavelength $\lambda = 350 \ \textrm {nm}$; the plasma conditions are $T_e = 2 \ \textrm {keV}$ and $n_e = 0.23 n_{c}$ ($n_{c}$ being the critical density). The growth rates are normalised to the maximal growth rate corresponding to each configuration. The left-hand panel represents the monochromatic case by solving (3.2) so it is normalised to $\gamma _{\textrm {max}} = \gamma _0 = \tfrac 14 k_0 v_{\textrm {os}}$. The middle and right-hand panels show the analytical broadband solution $\gamma _{\boldsymbol {K}}$ and are thus normalised to $\gamma _{\textrm {max}} = \gamma _{{L}} = 2 \gamma _0^2/\Delta \omega$.

The analytical results presented so far have relied upon the assumption that the bandwidth is large compared with the growth rate $\Delta \omega \gg \gamma _0$, which was used in the evaluation of the integral $\mathscr {I}$ in (3.9). To explore the growth rates of the instability in the intermediate regime $\Delta \omega \sim \gamma _0$, the integral is evaluated numerically for finite $\gamma$. Figure 3 shows the growth rate calculated along the lower branch of the TPD hyperbolas as a function of the magnitude of the wavenumber $K$. One sees that even at the fairly low bandwidth of $5 \, \textrm {THz}$ ($\Delta \omega / \gamma _0 = 2.8$) and at intensity $I_{{L}} = 10^{15} \, \textrm {W}\, \textrm {cm}^{-2}$, the growth rate is half its monochromatic value. It is also approximately a third for $\Delta \omega /2\pi = 10 \textrm {THz}$ (corresponding to the intrinsic bandwidth of an argon fluoride laser). As shown below, laser wavelength does not affect the peak values of the growth rate and therefore the results just quoted are unchanged by it. Decreasing the wavelength though does reduce the broadening effect in the plasma wave spectrum shown in figure 2.

Figure 3. Numerically evaluated growth rates $\gamma$ in the intermediate-bandwidth regime calculated along the lower branch of the TPD hyperbolas defined by $K_{\perp }^2 = K_{\parallel }(K_{\parallel } - k_0)$. The inset shows the middle panel of figure 2 along with the aforementioned hyperbola as represented by the red dashed line. The plasma conditions and laser parameters are the same as those described in figure 2. The growth rate is normalised to the monochromatic one $\gamma _0$, and the horizontal axis represents the magnitude of the wavenumber $K = \sqrt {K_\parallel ^2 + K_{\perp }^2}$. The circles represent the estimate of the maximum growth rate as given by (3.14).

In addition, it is possible to calculate the peak of the growth rates analytically in this regime. This is done by ignoring the thermal correction in the dispersion functions and assuming the wavenumber $\boldsymbol {K}$ is along the TPD hyperbola, leading to an expression for the maximum growth rate of the form

(3.13)\begin{equation} \gamma_{\text{max}} = \gamma_0^2 \int \frac{2 \omega_{\text{pe}} \,G(\omega_{k_{{\parallel}}})}{(\omega_{k_{{\parallel}}} - 2 \omega_{\text{pe}} - \textsf{i} \gamma_{\text{max}})(\omega_{k_{{\parallel}}} - \textsf{i} \gamma_{\text{max}})} \, \text{d} \omega_{k_{{\parallel}}}. \end{equation}

For a Lorentzian spectrum centred at $\omega _0 = 2 \omega _{\textrm {pe}}$ the integral can be evaluated exactly, leading to

(3.14)\begin{equation} \gamma_{\text{max}} = \tfrac14 ( \sqrt{16 \gamma_0^2 + \Delta \omega^2} - \Delta \omega). \end{equation}

This reproduces the two relevant limits: for $\Delta \omega \to 0, \gamma _{\textrm {max}} \to \gamma _0$; and for $\Delta \omega \to \infty, \gamma _{\textrm {max}} \to \gamma _{{L}} = 2\gamma _0^2/\Delta \omega$. Since $\gamma _0$ does not depend on laser wavelength, then $\gamma _{\textrm {max}}$ does not either. The circles in figure 3 represent $\gamma _{\textrm {max}}$ as given by the result above, showing excellent agreement with the numerically calculated growth rate.

3.3 Well-separated discrete spectral lines

Another implication concerns the extent to which a power spectrum consisting of $N$ well-separated sharp spectral lines spread over a bandwidth $\Delta \omega$, such as that shown in figure 4(a), approximates the instability reduction properties of a continuous power spectrum (Bodner Reference Bodner2023). For sufficiently low number of spectral lines, such that only a single line is contained within the instability resonance, $\Delta \omega /(N-1) > \gamma _0$, following the generalised dispersion relation presented here, the instability growth rate will be $\gamma = \gamma _0/\sqrt {N}$. From the aforementioned inequality, this reduction will be somewhat lower than the expected $\pi \gamma _0^2/\Delta \omega$ in the continuous case, but still significant. For example for $N=10$, the reduction is similar to that of $10 \ \textrm {THz}$ continuous bandwidth (see figure 3). At first glance this reduction cannot be explained by the coherence properties of the power spectrum, since formally if each line is delta-shaped, the coherence time is infinite.

Figure 4. (a) A power spectrum $G(\omega )$ consisting of $N=7$ well-separated spectral lines, centred around $\omega _0$ and spread over a bandwidth of $\Delta \omega$. Their thickness is finite for the purposes of visualisation. (b) The normalised correlation function $|\varGamma (\tau )|^2$ as a function of the time separation $\tau$ of a set of $N$ delta-shaped spectral lines spread over a bandwidth $\Delta \omega$.

To investigate this, one follows Goodman (Reference Goodman2015). Consider a power spectrum containing $N$ discrete delta-shaped spectral lines, uniformly spread over a bandwidth of $\Delta \omega$, and centred around $\omega _0$ (figure 4a):

(3.15)\begin{equation} G(\omega) = \frac{1}{N} \sum_{j=0}^{N-1} \delta \left( \omega - \omega_{0} - \frac{\Delta \omega}{2} + \frac{\Delta \omega}{N-1} j \right). \end{equation}

This is the power spectrum of the complex analytic signal $Y(t)$ where the actual signal is $y(t) = \textrm {Re} Y(t)$. By the Wiener–Khinchin theorem, the correlation function $C(\tau ) = \overline {Y^*(t) Y(t + \tau )}$ is given by the (inverse) Fourier transform of $G(\omega )$:

(3.16)\begin{equation} C(\tau) = \frac{1}{2\pi} \int G(\omega) \,{\rm e}^{- \textsf{i} \omega t} \,\text{d} t. \end{equation}

Defining the normalised correlation function $\varGamma (\tau ) = C(\tau )/C(0)$, the coherence time for a random process is given by

(3.17)\begin{equation} \tau_c = \int_{-\infty}^{\infty} |\varGamma(\tau)|^2 \,\text{d} \tau. \end{equation}

It can be shown that the correlation function for the discrete power spectrum $G(\omega )$ is

(3.18)\begin{equation} |\varGamma(\tau)|^2 = \frac{1}{N^2} \csc^2 \left( \frac{\Delta \omega \tau}{2(N-1)} \right) \sin^2 \left( \frac{N \Delta \omega \tau}{2(N-1)}\right). \end{equation}

Formally, the integral of this will always diverge for any finite $N$. In the limit of $N \to \infty$ one recovers the normalised correlation function for a continuous top hat spectrum which has a finite coherence time:

(3.19)\begin{equation} |\varGamma_{\text{TH}}(\tau)|^{2} = \text{sinc}^{2} (\Delta \omega \tau/2). \end{equation}

Figure 4(b) shows what $|\varGamma (\tau )|^2$ looks like for different numbers of spectral lines. One sees that for small $\tau$ the correlation functions closely resemble the sinc profile expected in the continuous case. In contrast, the discrete case exhibits correlations for larger $\tau$, and the time scale over which these correlations occur increases with the number of spectral lines $N$. The $k$th peak occurs at time delay $T_k = 2(N-1) \pi k / \Delta \omega$. If $T_1$, the time at which the first peak occurs, is much larger than the typical time scale of interest (such as $\gamma _0^{-1}$) then the coherence properties of the continuous spectrum are well approximated by the discrete one. Using $T_1 \gg \gamma _0^{-1}$ translates to a condition on the number of spectral lines:

(3.20)\begin{equation} N \gg \frac{1}{2\pi} \frac{\Delta \omega}{\gamma_0}. \end{equation}

For a laser intensity of $10^{15} \, \textrm {W} \, \textrm {cm}^{-2}$ at a wavelength of $350 \textrm {nm}$ and bandwidth of $\Delta \omega /\omega _0 = 1\,\%$, the right-hand side is ${\simeq } 1$. This suggests that even a low number of discrete spectral lines such as $N = 10$ might be a good approximation to the coherence properties of the continuous spectrum, as far as their effect on the instability is concerned.

4 Summary

In this paper, a novel statistical theory of the broadband TPD instability has been presented. It has been used to derive a dispersion relation in uniform plasma that is valid under laser electromagnetic fields with arbitrary power spectra. To achieve this, recent developments in inhomogeneous turbulence have been applied in order to derive the statistical closure required to treat the problem. The new dispersion relation is then applied to the study of the instability in a few cases of interest. For a single temporally incoherent laser beam, in the limit of large bandwidth $\Delta \omega \gg \gamma _0$, the well-known reduction of the peak of the growth rate by a factor of approximately $\gamma _0/\Delta \omega$ is confirmed, but it is also shown that the exact amount depends on the shape of the power spectrum. Namely, for a Lorentzian power spectrum the peak growth rate is $\gamma _{{L}} = 2 \gamma _0^2/\Delta \omega$, and for a flat (top-hat) spectrum it is $\gamma _{\textrm {TH}} = \pi \gamma _0^2/\Delta \omega$. This difference is less than what the difference in the coherence times of the two laser beams would imply. In addition, it has been shown that the laser bandwidth significantly broadens the range of wavenumbers excited by TPD. This implies that bandwidth favours the absolute instability in regions further away from the quarter critical density by allowing the $|\boldsymbol {K}| \to 0$ modes to be excited there too. The intermediate-bandwidth regime is explored numerically and it is shown that the growth rate is reduced to half its value for laser intensities of $10^{15} \, \textrm {W}\, \textrm {cm}^{-2}$ and relatively modest bandwidths of $5 \ \textrm {THz}$. A power spectrum consisting of $N$ discrete lines spread over a bandwidth of $\Delta \omega$ is also studied. It is shown that for a low number of spectral lines, the reduction in the peak growth rate is $1/\sqrt {N}$, which is somewhat smaller than in the continuous case but still significant.

There some natural questions which arise as a consequence of this work. Firstly, can one model transversely incoherent laser fields such as those produced by a random phase plate (RPP), or a combination of transverse and temporal incoherence such as when a phase plate is combined with smoothing by spectral dispersion (SSD), as well as in the induced spatial coherence (ISI) scheme? Formally transverse incoherence can be straightforwardly incorporated by simply allowing the Wigner function (equivalently, the laser power spectrum) to have some finite angular spread in wavenumber. But since two laser fields which can have the same power spectrum can nonetheless still be very different, care should be taken when applying the generalised dispersion relation. The RPP field is such that it creates a speckle pattern at the focus which may cause the instability to be strongly localised within speckles with many times the average intensity (Follett et al. Reference Follett, Wen, Froula, Turnbull and Palastro2022). Such intermittent behaviour may break the Gaussianity assumption made in this work. Adding temporal incoherence by SSD or by using naturally broad-bandwidth lasers such as the excimer ones used in the ISI scheme causes the speckle pattern to move which may help smooth out this behaviour.

Secondly, the presence of a density gradient is primarily responsible for determining whether the dominant instability at the quarter critical density is the TPD or Raman scattering, with longer, ignition-relevant scale lengths favouring the Raman instability (Rosenberg et al. Reference Rosenberg, Solodov, Myatt, Seka, Michel, Hohenberger, Short, Epstein, Regan, Campbell, Chapman, Goyon, Ralph, Barrios, Moody and Bates2018). Therefore extending the present analysis to study the effects of laser bandwidth on the absolute instability in the presence of a density gradient is an important problem. The approach of Liu & Rosenbluth (Reference Liu and Rosenbluth1976) and Simon et al. (Reference Simon, Short, Williams and Dewandre1983) for a plane-wave laser field uses some rather specific mathematical manipulations to calculate the absolute threshold, for which it is not immediately clear how they might be carried out within the framework presented in this paper. So this remains an interesting open problem.