2 Approach based on partial-differential equations

In this section, we present a formulation that is based on the partial-differential equations (PDE) governing the plasma motion. We assume that, to the zeroth order in $\unicode[STIX]{x1D700}$ , the plasma (including the photon content) is stationary and homogeneous. We also assume that the ions are motionless, that the electrons can be modelled as a fluid, and that the $O(\unicode[STIX]{x1D700}^{0})$ velocity of the electron fluid is zero. We also assume that the electrons are collisionless and non-magnetized. Hence, the wave dynamics is described as follows.

Consider the electron continuity equation

(2.1) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}n+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(n\boldsymbol{v})=0, & & \displaystyle\end{eqnarray}$$

where $n$ is the electron density, and $\boldsymbol{v}$ is the electron flow velocity. After the linearization, equation (2.1) becomes

(2.2) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}{\tilde{n}}+n_{0}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\tilde{\boldsymbol{v}}=0. & & \displaystyle\end{eqnarray}$$

(We use subscript 0 to denote $O(\unicode[STIX]{x1D700}^{0})$ quantities and tilde to denote $O(\unicode[STIX]{x1D700}^{1})$ quantities.) The velocity $\tilde{\boldsymbol{v}}$ is found from the electron momentum equation

(2.3) $$\begin{eqnarray}\displaystyle mn(\unicode[STIX]{x2202}_{t}+\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{v}=ne\boldsymbol{E}-\unicode[STIX]{x1D735}P+\boldsymbol{\unicode[STIX]{x1D702}}, & & \displaystyle\end{eqnarray}$$

where $m$ and $e<0$ are the electron mass and charge, $P$ is the pressure, and $\boldsymbol{\unicode[STIX]{x1D702}}$ is the ponderomotive force density (averaged over the EM-field oscillations yet not over the LW oscillations). After the linearization, equation (2.3) becomes

(2.4) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}\tilde{\boldsymbol{v}}=\frac{e}{m}\tilde{\boldsymbol{E}}-\frac{\unicode[STIX]{x1D735}\tilde{P}}{mn_{0}}+\frac{\tilde{\boldsymbol{\unicode[STIX]{x1D702}}}}{mn_{0}}. & & \displaystyle\end{eqnarray}$$

The electron gas is considered adiabatic, so $\tilde{P}=3mv_{T}^{2}{\tilde{n}}$ , where the constant $v_{T}$ is the unperturbed thermal speed of electrons. (See, e.g. appendix B.2 in Dodin, Geyko & Fisch (Reference Dodin, Geyko and Fisch2009), particularly (B21) and the references cited therein.) Hence, we obtain

(2.5) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}\tilde{\boldsymbol{v}}=\frac{e}{m}\,\tilde{\boldsymbol{E}}-3v_{T}^{2}\,\frac{\unicode[STIX]{x1D735}{\tilde{n}}}{n_{0}}+\frac{\tilde{\boldsymbol{\unicode[STIX]{x1D702}}}}{mn_{0}}. & & \displaystyle\end{eqnarray}$$

Substituting this into (2.2) gives $\unicode[STIX]{x2202}_{t}^{2}{\tilde{n}}=-n_{0}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x2202}_{t}\tilde{\boldsymbol{v}}$ , or

(2.6) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}^{2}{\tilde{n}}=-\frac{en_{0}}{m}\,\unicode[STIX]{x1D735}\boldsymbol{\cdot }\tilde{\boldsymbol{E}}+3v_{T}^{2}\unicode[STIX]{x1D6FB}^{2}{\tilde{n}}-\frac{\unicode[STIX]{x1D735}\boldsymbol{\cdot }\tilde{\boldsymbol{\unicode[STIX]{x1D702}}}}{m}. & & \displaystyle\end{eqnarray}$$

Using Gauss’s law, $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\tilde{\boldsymbol{E}}=4\unicode[STIX]{x03C0}e{\tilde{n}}$ , we then obtain

(2.7) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}^{2}{\tilde{n}}+\unicode[STIX]{x1D714}_{p,0}^{2}{\tilde{n}}-3v_{T}^{2}\unicode[STIX]{x1D6FB}^{2}{\tilde{n}}=-\frac{\unicode[STIX]{x1D735}\boldsymbol{\cdot }\tilde{\boldsymbol{\unicode[STIX]{x1D702}}}}{m}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D714}_{p}^{2}\doteq (4\unicode[STIX]{x03C0}ne^{2}/m)^{1/2}$ . (We use the symbol $\doteq$ to denote definitions.) Assume ${\tilde{n}}=\text{Re}\,(n_{c}\text{e}^{-\text{i}\unicode[STIX]{x1D6FA}t+\text{i}\boldsymbol{K}\boldsymbol{\cdot }\boldsymbol{x}})$ . A similar notation will also be assumed for other ‘tilded’ quantities; e.g.  $\tilde{\boldsymbol{\unicode[STIX]{x1D702}}}$ is real and $\tilde{\boldsymbol{\unicode[STIX]{x1D702}}}_{c}$ is complex. Then,

(2.8) $$\begin{eqnarray}\displaystyle [\unicode[STIX]{x1D6FA}^{2}-\unicode[STIX]{x1D6FA}_{0}^{2}(\boldsymbol{K})]{\tilde{n}}_{c}=\text{i}\boldsymbol{K}\boldsymbol{\cdot }\tilde{\boldsymbol{\unicode[STIX]{x1D702}}}_{c}/m, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FA}_{0}(\boldsymbol{K})=(\unicode[STIX]{x1D714}_{p,0}^{2}+3K^{2}v_{T}^{2})^{1/2}$ is the LW frequency absent photons. Note that $Kv_{T}\ll \unicode[STIX]{x1D6FA}$ is implied because otherwise electrons cannot be considered adiabatic but rather must be described kinetically (Stix Reference Stix1992); hence,

(2.9) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FA}_{0}(\boldsymbol{K})\approx \unicode[STIX]{x1D714}_{p,0}(1+3K^{2}v_{T}^{2}/2\unicode[STIX]{x1D714}_{p,0}^{2}). & & \displaystyle\end{eqnarray}$$

In the case of broad-band EM radiation, the average $\boldsymbol{\unicode[STIX]{x1D702}}$ equals the sum of the ponderomotive forces produced by its individual quasimonochromatic constituents, i.e. travelling geometrical-optics (GO) waves with well-defined wave vectors $\boldsymbol{k}$ and the corresponding frequencies

(2.10) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}=(\unicode[STIX]{x1D714}_{p}^{2}+c^{2}k^{2})^{1/2}. & & \displaystyle\end{eqnarray}$$

(In order to distinguish frequencies and wave vectors of EM waves from those of LW, we denote the former as $\unicode[STIX]{x1D714}$ and $\boldsymbol{k}$ . As a reminder, the frequency and wave vector of the LW are denoted as $\unicode[STIX]{x1D6FA}$ and $\boldsymbol{K}$ .) Each such EM wave produces a per-electron average force $-\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F7}_{\boldsymbol{k}}$ , where $\unicode[STIX]{x1D6F7}_{\boldsymbol{k}}=e^{2}|\boldsymbol{E}_{c}|^{2}/(4m\unicode[STIX]{x1D714}^{2})$ is the ponderomotive potential, and $\boldsymbol{E}_{c}$ is the complex amplitude of the EM-wave electric field $\boldsymbol{E}$ , which may have any polarization (Gaponov & Miller Reference Gaponov and Miller1958). (More specifically, we adopt $\boldsymbol{E}=\text{Re}\,(\boldsymbol{E}_{c}\text{e}^{\text{i}\unicode[STIX]{x1D703}})$ , where $\unicode[STIX]{x1D703}$ is the wave rapid phase. Accordingly, $\unicode[STIX]{x1D714}\doteq -\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D703}$ and $\boldsymbol{k}\doteq \unicode[STIX]{x1D735}\unicode[STIX]{x1D703}$ .) Hence, the average force density is $\boldsymbol{\unicode[STIX]{x1D702}}_{\boldsymbol{k}}=-n\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F7}_{\boldsymbol{k}}$ . Let us also express this in terms of the wave action density ${\mathcal{I}}={\mathcal{E}}/\unicode[STIX]{x1D714}$ (i.e. the photon density times $\hbar$ ), where (Dodin & Fisch Reference Dodin and Fisch2012)

(2.11) $$\begin{eqnarray}\displaystyle {\mathcal{E}}=\frac{1}{16\unicode[STIX]{x03C0}\unicode[STIX]{x1D714}}\boldsymbol{E}_{c}^{\ast }\boldsymbol{\cdot }\unicode[STIX]{x2202}_{\unicode[STIX]{x1D714}}[\unicode[STIX]{x1D714}^{2}\unicode[STIX]{x1D750}(\unicode[STIX]{x1D714},\boldsymbol{k})]\boldsymbol{\cdot }\boldsymbol{E}_{c} & & \displaystyle\end{eqnarray}$$

is the wave energy density, and $\unicode[STIX]{x1D750}$ is the dielectric tensor. Using $\unicode[STIX]{x1D750}(\unicode[STIX]{x1D714},\boldsymbol{k})=1-\unicode[STIX]{x1D714}_{p}^{2}/\unicode[STIX]{x1D714}^{2}$ , we obtain ${\mathcal{E}}=|\boldsymbol{E}_{c}|^{2}/(8\unicode[STIX]{x03C0})$ . Then, irrespective of polarization, the average force density is given by the following expression:

(2.12) $$\begin{eqnarray}\displaystyle \boldsymbol{\unicode[STIX]{x1D702}}_{\boldsymbol{k}}=-\frac{\unicode[STIX]{x1D714}_{p}^{2}}{2}\,\unicode[STIX]{x1D735}\left(\frac{{\mathcal{I}}}{\unicode[STIX]{x1D714}}\right). & & \displaystyle\end{eqnarray}$$

As a side remark, note that the force $\boldsymbol{\unicode[STIX]{x1D702}}_{\boldsymbol{k}}$ induced by a modulation of an EM wave on a plasma is qualitatively different from the force induced by a modulation of a matter wave on a plasma. In the electrostatic limit considered here, a matter wave causes an electrostatic force that is entirely determined by the electron density modulation through Gauss’s law. In contrast, $\boldsymbol{\unicode[STIX]{x1D702}}_{\boldsymbol{k}}$ is determined by the modulation of both the photon density ( ${\mathcal{I}}/\hbar$ ) and photon energy ( $\hbar \unicode[STIX]{x1D714}$ ), so it can be non-zero even when the photon density is homogeneous. Considering that the photon Hamiltonian, or frequency (2.10), is very similar to the Hamiltonian of a relativistic electron, this difference may seem surprising; one could expect a one-to-one correspondence between electrostatic and ponderomotive forces. However, the correspondence is limited to free particles only. An electron has a constant rest mass and is coupled to the environment through the electrostatic potential. In contrast, a photon has a variable rest mass determined by $\unicode[STIX]{x1D714}_{p}$ and is coupled to the environment through modulations of this mass. Even in the limit $\unicode[STIX]{x1D714}_{p}\ll kc$ , the interaction Hamiltonian of a photon remains $k$ -dependent, so it cannot be represented as a potential energy in principle. This explains the difference between the corresponding forces. (As another side remark, this also explains the non-zero term (A 3) in the photon ponderomotive potential (A 2), which has no analogue in the electron ponderomotive potential, at least in the electrostatic gauge.)

The force density $\boldsymbol{\unicode[STIX]{x1D702}}$ produced by broad-band radiation can be written as $\boldsymbol{\unicode[STIX]{x1D702}}=\int \boldsymbol{\unicode[STIX]{x1D702}}_{\boldsymbol{k}}\,\text{d}^{3}k$ , where ${\mathcal{I}}$ is replaced with the phase-space action density $F$ summed over all polarization states. (Accordingly, the result holds for polarized and depolarized radiation equally. Also note that $F/\hbar$ can be understood as the phase-space photon probability distribution (Dodin Reference Dodin2014a ).) This gives

(2.13) $$\begin{eqnarray}\displaystyle \boldsymbol{\unicode[STIX]{x1D702}}(t,\boldsymbol{x})=-\frac{\unicode[STIX]{x1D714}_{p}^{2}}{2}\,\unicode[STIX]{x1D735}\int \frac{F(t,\boldsymbol{x},\boldsymbol{k})}{\unicode[STIX]{x1D714}(t,\boldsymbol{x},\boldsymbol{k})}\,\text{d}^{3}k. & & \displaystyle\end{eqnarray}$$

Equation (2.13) is in agreement with the formula reported previously (Bingham et al. Reference Bingham, Mendonça and Dawson1997; Mendonça Reference Mendonça2000), at least up to a factor of two. However, in this work, we report a different linearization, which is as follows:

(2.14) $$\begin{eqnarray}\displaystyle \tilde{\boldsymbol{\unicode[STIX]{x1D702}}}_{c}=-\frac{\text{i}}{2}\,\boldsymbol{K}\unicode[STIX]{x1D714}_{p,0}^{2}\int \left(\frac{\tilde{F}_{c}}{\unicode[STIX]{x1D714}_{0}}-\frac{\tilde{\unicode[STIX]{x1D714}}_{c}F_{0}}{\unicode[STIX]{x1D714}_{0}^{2}}\right)\,\text{d}^{3}k. & & \displaystyle\end{eqnarray}$$

Here, $\unicode[STIX]{x1D714}_{0}\doteq (\unicode[STIX]{x1D714}_{p,0}^{2}+c^{2}k^{2})^{1/2}$ is the unperturbed frequency of the EM wave, and

(2.15) $$\begin{eqnarray}\displaystyle \tilde{\unicode[STIX]{x1D714}}_{c}=\frac{\unicode[STIX]{x1D714}_{p,0}^{2}}{2\unicode[STIX]{x1D714}_{0}}\frac{{\tilde{n}}_{c}}{n_{0}} & & \displaystyle\end{eqnarray}$$

is the perturbation on the EM-wave frequency due to the plasma density variations caused by the LW. The second term under the integral in (2.14) is specific to the photon–plasma interaction and has no direct analogue in the electron–plasma interaction for reasons discussed in the previous paragraph.

The perturbation of the photon distribution $\tilde{F}_{c}$ is obtained from the wave kinetic equation

(2.16) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}F+\unicode[STIX]{x2202}_{\boldsymbol{k}}\unicode[STIX]{x1D714}\boldsymbol{\cdot }\unicode[STIX]{x1D735}F-\unicode[STIX]{x2202}_{\boldsymbol{x}}\unicode[STIX]{x1D714}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\boldsymbol{k}}F=0, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}(t,\boldsymbol{x},\boldsymbol{k})$ and $\unicode[STIX]{x2202}_{\boldsymbol{k}}\unicode[STIX]{x1D714}$ is understood as the group velocity. Since the unperturbed plasma is considered homogeneous, linearizing equation (2.16) gives

(2.17) $$\begin{eqnarray}\displaystyle \tilde{F}_{c}=-\frac{\tilde{\unicode[STIX]{x1D714}}_{c}\boldsymbol{K}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\boldsymbol{k}}F_{0}}{\unicode[STIX]{x1D6FA}-\boldsymbol{K}\boldsymbol{\cdot }\boldsymbol{v}_{\ast }}, & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{v}_{\ast }\doteq c\boldsymbol{k}/\unicode[STIX]{x1D714}_{0}$ is the unperturbed group velocity. Inserting equations (2.15) and (2.17) into (2.14), we obtain

(2.18) $$\begin{eqnarray}\displaystyle \tilde{\boldsymbol{\unicode[STIX]{x1D702}}}_{c}=\frac{\text{i}}{4}\left(\frac{{\tilde{n}}_{c}}{n_{0}}\right)\boldsymbol{K}\unicode[STIX]{x1D714}_{p,0}^{4}Q(\unicode[STIX]{x1D6FA},\boldsymbol{K}), & & \displaystyle\end{eqnarray}$$

where we introduced the following function:

(2.19) $$\begin{eqnarray}\displaystyle Q(\unicode[STIX]{x1D6FA},\boldsymbol{K})\doteq \int \left[\frac{\boldsymbol{K}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\boldsymbol{k}}F_{0}(\boldsymbol{k})}{(\unicode[STIX]{x1D6FA}-\boldsymbol{K}\boldsymbol{\cdot }\boldsymbol{v}_{\ast })\unicode[STIX]{x1D714}_{0}^{2}(k)}+\frac{F_{0}(\boldsymbol{k})}{\unicode[STIX]{x1D714}_{0}^{3}(k)}\right]\,\text{d}^{3}k. & & \displaystyle\end{eqnarray}$$

Note that the second term in the brackets is due to the second term in (2.14).

By substituting (2.18) into (2.8), we obtain

(2.20) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FA}^{2}-\unicode[STIX]{x1D6FA}_{0}^{2}(\boldsymbol{K})=-\frac{\unicode[STIX]{x1D714}_{p,0}^{4}K^{2}}{4mn_{0}}\,Q(\unicode[STIX]{x1D6FA},\boldsymbol{K}). & & \displaystyle\end{eqnarray}$$

Assuming that $\unicode[STIX]{x1D6FA}$ is close to $\unicode[STIX]{x1D6FA}_{0}\approx \unicode[STIX]{x1D714}_{p,0}$ , we can also simplify the left-hand side here as follows:

(2.21) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FA}^{2}-\unicode[STIX]{x1D6FA}_{0}^{2}(\boldsymbol{K})\approx 2\unicode[STIX]{x1D714}_{p,0}[\unicode[STIX]{x1D6FA}-\unicode[STIX]{x1D6FA}_{0}(\boldsymbol{K})]. & & \displaystyle\end{eqnarray}$$

Then, the dispersion relation becomes

(2.22) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FA}\approx \unicode[STIX]{x1D6FA}_{0}(\boldsymbol{K})-\frac{\unicode[STIX]{x1D714}_{p,0}^{3}K^{2}}{8mn_{0}}Q(\unicode[STIX]{x1D6FA},\boldsymbol{K}). & & \displaystyle\end{eqnarray}$$

The second term on the right-hand side is the modification of the LW dispersion relation caused by the photon gas. Equation (2.22) differs from the corresponding relation reported previously (Bingham et al. Reference Bingham, Mendonça and Dawson1997; Mendonça Reference Mendonça2000) in the following aspects: (a) the order-one numerical coefficient in front of the integral is different, and most importantly, the second term in the integrand in our expression (2.19) for $Q$ is new. Notably, such terms are missed also in the general method reported by Tsytovich (Reference Tsytovich1970) for calculating the nonlinear plasma dispersion. The wave-scattering paradigm that underlies this method assumes that the interaction between each pair of waves can be modelled as instantaneous, so the effect of the adiabatic frequency shift $\unicode[STIX]{x1D714}-\unicode[STIX]{x1D714}_{0}$ cannot be captured.

In order to estimate the photon contribution, suppose for clarity that $\unicode[STIX]{x1D714}\sim kc$ and $k\sim K$ . (Under the assumptions adopted in the present section, this regime is accessible only marginally, but a more general theory given in § 3 leads to similar estimates.) Then, the ratio of the two terms in the right-hand side of (2.22) is roughly

(2.23) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D714}_{p,0}^{3}K^{2}}{mn_{0}\unicode[STIX]{x1D6FA}_{0}}Q\sim \left(\frac{eE_{c}}{mc\unicode[STIX]{x1D714}}\right)^{2}\equiv a^{2}, & & \displaystyle\end{eqnarray}$$

where we assumed $Q\sim \unicode[STIX]{x1D714}^{-3}\int F_{0}\,\text{d}^{3}k\sim {\mathcal{I}}/\unicode[STIX]{x1D714}^{3}\sim |E_{c}|^{2}/(8\unicode[STIX]{x03C0}\unicode[STIX]{x1D714}^{4})$ . Note that $a$ is the amplitude of the EM-driven momentum oscillations in units $mc$ . Thus, in the non-relativistic limit assumed here, one has $a\ll 1$ , so the photon contribution is small. Nevertheless, photons can have an important effect on the LW stability. For example, when resonant photons are present, the integrand in (2.19) has a pole on the real axis, and the integral must be taken along the Landau contour (Stix Reference Stix1992). Then, $\unicode[STIX]{x1D6FA}$ is complex, which signifies dissipation (positive or negative) of LW on photons. This effect is known as photon Landau damping (Bingham et al. Reference Bingham, Mendonça and Dawson1997; Mendonça Reference Mendonça2000) and has been demonstrated experimentally, albeit in a solid-state medium rather than plasma (Dylov & Fleischer Reference Dylov and Fleischer2008). Also note that LW are considered here only as a simple example, and the effect of photons on the dispersion of other waves can be more substantial.

3 Polarizability-based approach

Although the calculations of the ponderomotive forces are relatively straightforward in the situation considered in this paper, the problem can be much harder when $\unicode[STIX]{x1D6F7}_{\boldsymbol{k}}$ is velocity-dependent, i.e. when kinetic effects are essential (Dodin & Fisch Reference Dodin and Fisch2008a ,Reference Dodin and Fisch b ; Dodin Reference Dodin2014b ). Thus, it would be advantageous to develop a formulation of the modulational dynamics that avoids this step altogether. Below, we propose such formulation that utilizes the photon-polarizability concept (Ruiz & Dodin Reference Ruiz and Dodin2016). Within this approach, ponderomotive forces do not need to be considered, and a more general dispersion relation is obtained.

3.1 General theory

The dispersion relation of electrostatic oscillations in plasma with dielectric tensor $\unicode[STIX]{x1D750}$ is given by

(3.1) $$\begin{eqnarray}\displaystyle \boldsymbol{e}_{\boldsymbol{K}}\boldsymbol{\cdot }\unicode[STIX]{x1D750}(\unicode[STIX]{x1D6FA},\boldsymbol{K})\boldsymbol{\cdot }\boldsymbol{e}_{\boldsymbol{K}}=0, & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{e}_{\boldsymbol{K}}\doteq \boldsymbol{K}/K$ is the unit vector along the LW wave vector $\boldsymbol{K}$ and $\unicode[STIX]{x1D6FA}$ is the LW frequency. Consider a plasma whose dielectric tensor is some $\unicode[STIX]{x1D750}_{0}$ plus the contribution from the photon gas. The latter contribution is the photon susceptibility, which can be written as follows:

(3.2) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D74C}_{ph}(\unicode[STIX]{x1D6FA},\boldsymbol{K})=4\unicode[STIX]{x03C0}\int \unicode[STIX]{x1D736}_{ph}(\unicode[STIX]{x1D6FA},\boldsymbol{K},\boldsymbol{k})f_{ph}(\boldsymbol{k})\,\text{d}^{3}k. & & \displaystyle\end{eqnarray}$$

Here, $f_{ph}$ is the unperturbed photon distribution, and $\unicode[STIX]{x1D736}_{ph}$ is the polarizability of a single photon. As shown previously (Ruiz & Dodin Reference Ruiz and Dodin2016),

(3.3) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D736}_{ph}=\frac{\hbar e^{2}K^{2}\unicode[STIX]{x1D6EF}}{4m^{2}\unicode[STIX]{x1D714}_{0}^{3}}\,\boldsymbol{e}_{\boldsymbol{K}}\boldsymbol{e}_{\boldsymbol{K}}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6EF}$ is a dimensionless coefficient given by

(3.4) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6EF}\doteq \frac{\unicode[STIX]{x1D6FA}^{2}-c^{2}K^{2}}{(\unicode[STIX]{x1D6FA}-\boldsymbol{K}\boldsymbol{\cdot }\boldsymbol{v}_{\ast })^{2}-(\unicode[STIX]{x1D6FA}^{2}-c^{2}K^{2})^{2}/4\unicode[STIX]{x1D714}_{0}^{2}(k)}, & & \displaystyle\end{eqnarray}$$

or, equivalently,

(3.5) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6EF}=-\mathop{\sum }_{\unicode[STIX]{x1D70E}=\pm 1}\frac{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D714}_{0}(k)}{R_{\unicode[STIX]{x1D70E}}(\unicode[STIX]{x1D6FA},\boldsymbol{K},\boldsymbol{k})}, & \displaystyle\end{eqnarray}$$

(3.6) $$\begin{eqnarray}\displaystyle & \displaystyle R_{\unicode[STIX]{x1D70E}}(\unicode[STIX]{x1D6FA},\boldsymbol{K},\boldsymbol{k})\doteq \unicode[STIX]{x1D6FA}-\boldsymbol{K}\boldsymbol{\cdot }\boldsymbol{v}_{\ast }(\boldsymbol{k})+\unicode[STIX]{x1D70E}\,\frac{\unicode[STIX]{x1D6FA}^{2}-c^{2}K^{2}}{2\unicode[STIX]{x1D714}_{0}(k)}. & \displaystyle\end{eqnarray}$$

Thus, equation (3.1) can be expressed as follows:

(3.7) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D716}_{0}(\unicode[STIX]{x1D6FA},\boldsymbol{K})+\unicode[STIX]{x1D712}_{ph}(\unicode[STIX]{x1D6FA},\boldsymbol{K})=0. & & \displaystyle\end{eqnarray}$$

Here, $\unicode[STIX]{x1D716}_{0}\doteq \boldsymbol{e}_{\boldsymbol{K}}\boldsymbol{\cdot }\unicode[STIX]{x1D750}_{0}(\unicode[STIX]{x1D6FA},\boldsymbol{K})\boldsymbol{\cdot }\boldsymbol{e}_{\boldsymbol{K}}$ and $\unicode[STIX]{x1D712}_{ph}(\unicode[STIX]{x1D6FA},\boldsymbol{K})\doteq \boldsymbol{e}_{\boldsymbol{K}}\boldsymbol{\cdot }\unicode[STIX]{x1D74C}_{ph}(\unicode[STIX]{x1D6FA},\boldsymbol{K})\boldsymbol{\cdot }\boldsymbol{e}_{\boldsymbol{K}}$ or, explicitly,

(3.8) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D712}_{ph}(\unicode[STIX]{x1D6FA},\boldsymbol{K})=-\frac{\unicode[STIX]{x1D714}_{p,0}^{2}K^{2}}{4mn_{0}}\mathop{\sum }_{\unicode[STIX]{x1D70E}=\pm 1}\int \frac{\unicode[STIX]{x1D70E}F_{0}(\boldsymbol{k})}{\unicode[STIX]{x1D714}_{0}^{2}(k)R_{\unicode[STIX]{x1D70E}}(\unicode[STIX]{x1D6FA},\boldsymbol{K},\boldsymbol{k})}\,\text{d}^{3}k, & & \displaystyle\end{eqnarray}$$

where $F_{0}\doteq \hbar f_{ph}$ is the photon action density, which is a classical quantity. Accordingly, equation (3.7) becomes

(3.9) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D716}_{0}(\unicode[STIX]{x1D6FA},\boldsymbol{K})-\frac{\unicode[STIX]{x1D714}_{p,0}^{2}K^{2}}{4mn_{0}}\mathop{\sum }_{\unicode[STIX]{x1D70E}=\pm 1}\int \frac{\unicode[STIX]{x1D70E}F_{0}(\boldsymbol{k})}{\unicode[STIX]{x1D714}_{0}^{2}(k)R_{\unicode[STIX]{x1D70E}}(\unicode[STIX]{x1D6FA},\boldsymbol{K},\boldsymbol{k})}\,\text{d}^{3}k=0. & & \displaystyle\end{eqnarray}$$

Compared to (2.22) that was obtained within a PDE-based approach, equation (3.9) is more general. It allows for $k\sim K$ and does not assume any specific model of the background plasma; i.e. no assumptions regarding $\unicode[STIX]{x1D750}_{0}$ are made. Moreover, the derivation applies as is to the case where the background plasma is weakly non-stationary and (or) weakly inhomogeneous. (For more details, see Ruiz & Dodin (Reference Ruiz and Dodin2016).) Also importantly, the same approach is readily extended to describe modulational dynamics of other waves too, as will be reported separately.

3.2 Small- $K$ limit

Finally, let us show how (3.9) reduces to (2.22) under the additional assumptions adopted in § 2. First, assume the GO approximation for photons, namely, $K\ll k$ . This implies (Dodin & Fisch Reference Dodin and Fisch2014; Ruiz & Dodin Reference Ruiz and Dodin2016)

(3.10) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6EF}(\unicode[STIX]{x1D6FA},\boldsymbol{K},\boldsymbol{k})\approx \frac{\unicode[STIX]{x1D6FA}^{2}-c^{2}K^{2}}{(\unicode[STIX]{x1D6FA}-\boldsymbol{K}\boldsymbol{\cdot }\boldsymbol{v}_{\ast })^{2}}. & & \displaystyle\end{eqnarray}$$

Hence, the photon susceptibility becomes

(3.11) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D712}_{ph}(\unicode[STIX]{x1D6FA},\boldsymbol{K})=\frac{\unicode[STIX]{x1D714}_{p,0}^{2}K^{2}}{4mn_{0}}\int \frac{\unicode[STIX]{x1D6FA}^{2}-c^{2}K^{2}}{(\unicode[STIX]{x1D6FA}-\boldsymbol{K}\boldsymbol{\cdot }\boldsymbol{v}_{\ast })^{2}\unicode[STIX]{x1D714}_{0}^{3}(k)}\,F_{0}(\boldsymbol{k})\,\text{d}^{3}k. & & \displaystyle\end{eqnarray}$$

As can be checked by a direct calculation,

(3.12) $$\begin{eqnarray}\displaystyle \boldsymbol{K}\boldsymbol{\cdot }\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\boldsymbol{k}}\left[\frac{1}{(\unicode[STIX]{x1D6FA}-\boldsymbol{K}\boldsymbol{\cdot }\boldsymbol{v}_{\ast })\unicode[STIX]{x1D714}_{0}^{2}}\right]=\frac{c^{2}K^{2}-\unicode[STIX]{x1D6FA}^{2}}{(\unicode[STIX]{x1D6FA}-\boldsymbol{K}\boldsymbol{\cdot }\boldsymbol{v}_{\ast })^{2}\unicode[STIX]{x1D714}_{0}^{3}}+\frac{1}{\unicode[STIX]{x1D714}_{0}^{3}}. & & \displaystyle\end{eqnarray}$$

(This non-intuitive step can be avoided as explained in appendix.) Thus, one can also express $\unicode[STIX]{x1D712}_{ph}$ equivalently as follows:

(3.13) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D712}_{ph}(\unicode[STIX]{x1D6FA},\boldsymbol{K})=\frac{\unicode[STIX]{x1D714}_{p,0}^{2}K^{2}}{4mn_{0}}Q(\unicode[STIX]{x1D6FA},\boldsymbol{K}), & & \displaystyle\end{eqnarray}$$

where $Q$ is given by (2.19). Second, assume that ions are motionless and electrons have a non-zero yet small enough thermal speed $v_{T}\ll \unicode[STIX]{x1D6FA}/K$ ; then (Stix Reference Stix1992),

(3.14) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D716}_{0}(\unicode[STIX]{x1D6FA},\boldsymbol{K})=1-\frac{\unicode[STIX]{x1D714}_{p,0}^{2}}{\unicode[STIX]{x1D6FA}^{2}}\left(1+\frac{3K^{2}v_{T}^{2}}{\unicode[STIX]{x1D714}_{p,0}^{2}}\right), & & \displaystyle\end{eqnarray}$$

and $\unicode[STIX]{x1D6FA}$ is close to the unperturbed linear frequency $\unicode[STIX]{x1D6FA}_{0}$ (2.9). This can be simplified to $\unicode[STIX]{x1D716}_{0}\approx 2[\unicode[STIX]{x1D6FA}-\unicode[STIX]{x1D6FA}_{0}(\boldsymbol{K})]/\unicode[STIX]{x1D714}_{p,0}$ . Hence, equation (3.7) immediately leads to (2.22).