**4. Volkov solution in a dielectric medium**

The mathematical approach to the situation where we consider plane wave solution in a medium is the same, only with the difference that the Lorentz condition must be replaced according to Schwinger et al. by the following one [14]:

$$
\partial\_{\mu}A^{\mu} = kA^{\prime} = (\mu\varepsilon - 1)(\eta\partial)(\eta A) = (\mu\varepsilon - 1)(\eta k)(\eta A^{\prime})\tag{25}
$$

with the specification *η<sup>µ</sup>* = (1, **0**) as the unit time-like vector in the rest frame of the medium [14].

For periodic potential *Aµ*, we then get from eq. (25) instead of *kA* = 0 the following equation:

$$kA = (\mu \varepsilon - 1)(\eta k)(\eta A). \tag{26}$$

Then, we get instead of eq. (14) the following equation for function *F*(*ϕ*):

$$2\mathrm{i}(kp)\mathrm{F}^{\prime} + [-2\mathrm{e}(pA) + \mathrm{e}^{2}A^{2} - \mathrm{i}\mathrm{e}(\gamma k)(\gamma A^{\prime}) - \mathrm{i}\mathrm{e}(\mu\varepsilon - 1)(\eta k)(\eta A^{\prime})]\mathrm{F} = 0. \tag{27}$$

The solution of the last equation is the solution of the linear equation of the form *y* + *Py* = 0, and it means it is of the form *<sup>y</sup>* <sup>=</sup> *<sup>C</sup>* exp(<sup>−</sup> *Pdx*), where C is some constant. So, we can write the solution as follows:

$$F = \exp\left\{-i\int\_0^{kx} \left[\frac{e}{(kp)}(pA) - \frac{e^2}{2(kp)}A^2\right]d\varphi + \frac{\varepsilon(\gamma k)(\gamma A)}{2(kp)} + \frac{e}{2(kp)}a\right\}\frac{u}{\sqrt{2p\_0}},\tag{28}$$

where

$$
\mathfrak{a} = (\mathfrak{\mu}\varepsilon - 1)(\mathfrak{\eta}k)(\mathfrak{\eta}A) \tag{29}
$$

The wave function *ψ* is then the modified wave function (18), which we can write in the form

$$\psi\_p = \left[1 + \sum\_{n=1}^{\infty} \left(\frac{e}{2(kp)}\right)^n (2a)^{n-1} (\gamma k)(\gamma A)\right] \frac{u}{\sqrt{2p\_0}} e^{iS} e^T \tag{30}$$

where

$$T = \frac{e}{2(kp)}(\mu\varepsilon - 1)(\eta k)(\eta A),\tag{31}$$

and where we used in the last formula the following relation:

$$[(\gamma k)(\gamma A)]^n = (2a)^{n-1}(\gamma k)(\gamma A). \tag{32}$$

So, we see that the influence of the medium on the Volkov solution is involved in exp(*T*), where *T* is given by eq. (31) and in the new term which involves the sum of the infinite number of coefficients.
