**5. Compton effect from Volkov solution**

We determine the Compton process in vacuum as a result of the Volkov solution of the Dirac equation. The Volkov solution involves not only the one-photon scattering but also the multiphoton scattering of photons on electron. In the time of the Compton experiment in 1922, the Volkov solution was not known, because the Dirac equation was published in 1928 and the Volkov solution in 1935.

Now, let us consider electromagnetic monochromatic plane wave which is polarized in circle. We write the four-potential in the form

$$A = a\_1 \cos \varphi + a\_2 \sin \varphi,\tag{33}$$

where the amplitudes *ai* are the same and mutually perpendicular, or

$$a\_1^2 = a\_2^2 = a^2, \quad a\_1 a\_2 = 0. \tag{34}$$

The Volkov solution for the standard vacuum situation is of the form

$$\psi\_p = \left\{ 1 + \left(\frac{e}{2(kp)}\right) \left[ (\gamma k)(\gamma a\_1) \cos \varphi + (\gamma k)(\gamma a\_2) \sin \varphi \right] \right\} \frac{u(p)}{\sqrt{2q\_0}} \quad \times$$

$$\exp\left\{ -ie \frac{(a\_1 p)}{(kp)} \sin \varphi + ie \frac{(a\_2 p)}{(kp)} \cos \varphi - iqx \right\}, \tag{35}$$

where

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

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

<sup>2</sup>*A*<sup>2</sup> <sup>−</sup> *ie*(*γk*)(*γ<sup>A</sup>*

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

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

(*pA*) <sup>−</sup> *<sup>e</sup>*<sup>2</sup>

 *e* 2(*kp*)

*<sup>T</sup>* <sup>=</sup> *<sup>e</sup>* 2(*kp*)

[(*γk*)(*γA*)]

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

2(*kp*)

*A*2 *dϕ* +

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

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

(2*α*)*n*−1(*γk*)(*γA*)

*<sup>n</sup>*

2*i*(*kp*)*F* + [−2*e*(*pA*) + *e*

 *e* (*kp*)

*ψ<sup>p</sup>* =

 1 + ∞ ∑ *n*=1

write the solution as follows:

118 Graphene - New Trends and Developments

 −*i kx* 0

*F* = exp

where

where

number of coefficients.

*kA* = (*µε* − 1)(*ηk*)(*ηA*). (26)

) − *ie*(*µε* − 1)(*ηk*)(*ηA*

*α* = (*µε* − 1)(*ηk*)(*ηA*) (29)

 *u* 2*p*<sup>0</sup> *e*

(*µε* − 1)(*ηk*)(*ηA*), (31)

*<sup>n</sup>* = (2*α*)*n*−1(*γk*)(*γA*). (32)

*e* 2(*kp*) *α u* 2*p*<sup>0</sup>

*e*(*γk*)(*γA*) <sup>2</sup>(*kp*) <sup>+</sup> )]*F* = 0. (27)

, (28)

*iSeT*, (30)

[14].

$$q^{\mu} = p^{\mu} - e^2 \frac{a^2}{2(kp)} k^{\mu}. \tag{36}$$

because it follows from eq. (24).

We know that the matrix element *M* corresponding to the emission of photon by electron in the electromagnetic field is as follows [12]:

$$S\_{fi} = -ie^2 \int d^4x \bar{\psi}\_{p'}(\gamma e'^\*) \psi\_p \frac{e^{i\mathbf{k}' \cdot \mathbf{x}}}{\sqrt{2\omega'}},\tag{37}$$

where *ψ<sup>p</sup>* is the electron wave function before the interaction of electron with the laser pulse and *ψ<sup>p</sup>* is the electron wave function after emission of photon with components *k<sup>µ</sup>* = (*ω* , **k** ). Symbol *e*∗ is the four polarization vector of emitted photon.

Then, we get the following linear combination in the matrix element:

$$e^{-i\alpha\_1 \sin \varphi + i\alpha\_2 \cos \varphi} \tag{38}$$

$$e^{-i\mathbf{a}\_1 \sin \varphi + i\mathbf{a}\_2 \cos \varphi} \cos \varphi \tag{39}$$

$$e^{-i\alpha\_1 \sin \varphi + i\alpha\_2 \cos \varphi} \sin \varphi. \tag{40}$$

where

$$\alpha\_1 = e \left( \frac{a\_1 p}{kp} - \frac{a\_1 p'}{kp'} \right),\tag{41}$$

and

$$a\_2 = e\left(\frac{a\_2 p}{kp} - \frac{a\_2 p'}{kp'}\right). \tag{42}$$

Now, we can expand the exponential function into the Fourier transformation where the coefficients of the expansion will be *Bs*, *B*1*s*, *B*2*s*. So we write

$$e^{-ia\_1\sin\varphi + ia\_2\cos\varphi} = e^{-i\sqrt{a\_1^2 + a\_2^2}\sin(\varphi - \varphi\_0)} = \sum\_{s=-\infty}^{\infty} B\_s e^{-is\varphi} \tag{43}$$

$$e^{-i\mu\_1 \sin \varphi + i\mu\_2 \cos \varphi} \cos \varphi = e^{-i\sqrt{a\_1^2 + a\_2^2} \sin(\varphi - \varphi\_0)} \cos \varphi = \sum\_{s=-\infty}^{\infty} B\_{1s} e^{-is\varphi} \tag{44}$$

$$e^{-i\alpha\_1 \sin \varphi + i\alpha\_2 \cos \varphi} \sin \varphi = e^{-i\sqrt{a\_1^2 + a\_2^2} \sin(\varphi - \varphi\_0)} \sin \varphi = \sum\_{s = -\infty}^{\infty} B\_{2s} e^{-is\varphi}. \tag{45}$$

The coefficients *Bs*, *B*1*s*, *B*2*<sup>s</sup>* can be expressed by means of the Bessel function as follows [12]:

$$B\_s = f\_s(z)e^{i s \varphi\_0} \tag{46}$$

$$B\_{1s} = \frac{1}{2} \left[ J\_{s+1}(z)e^{i(s+1)\varphi\_0} + J\_{s-1}(z)e^{i(s-1)\varphi\_0} \right] \tag{47}$$

$$B\_{2s} = \frac{1}{2i} \left[ J\_{s+1}(z)e^{i(s+1)\varphi\_0} - J\_{s-1}(z)e^{i(s-1)\varphi\_0} \right] \tag{48}$$

where the quantity *z* is defined in [12] through *α* components as follows:

$$z = \sqrt{\mathfrak{a}\_1^2 + \mathfrak{a}\_2^2} \tag{49}$$

and

*e*

*e*

*e*

*α*<sup>1</sup> = *e*

*α*<sup>2</sup> = *e*

−*i* √*α*<sup>2</sup> 1+*α*<sup>2</sup>

−*i* √*α*<sup>2</sup> 1+*α*<sup>2</sup>

−*i* √*α*<sup>2</sup> 1+*α*<sup>2</sup>

*Js*+1(*z*)*e*

The coefficients *Bs*, *B*1*s*, *B*2*<sup>s</sup>* can be expressed by means of the Bessel function as follows [12]:

*<sup>i</sup>*(*s*+1)*ϕ*<sup>0</sup> <sup>+</sup> *Js*−1(*z*)*<sup>e</sup>*

*Bs* = *Js*(*z*)*e*

coefficients of the expansion will be *Bs*, *B*1*s*, *B*2*s*. So we write

<sup>−</sup>*iα*<sup>1</sup> sin *<sup>ϕ</sup>*+*iα*<sup>2</sup> cos *<sup>ϕ</sup>* = *e*

<sup>−</sup>*iα*<sup>1</sup> sin *<sup>ϕ</sup>*+*iα*<sup>2</sup> cos *<sup>ϕ</sup>* cos *ϕ* = *e*

<sup>−</sup>*iα*<sup>1</sup> sin *<sup>ϕ</sup>*+*iα*<sup>2</sup> cos *<sup>ϕ</sup>* sin *ϕ* = *e*

*<sup>B</sup>*1*<sup>s</sup>* <sup>=</sup> <sup>1</sup> 2 

*e*

*e*

120 Graphene - New Trends and Developments

*e*

*a*<sup>1</sup> *p*

*a*<sup>2</sup> *p*

Now, we can expand the exponential function into the Fourier transformation where the

*kp* <sup>−</sup> *<sup>a</sup>*<sup>1</sup> *<sup>p</sup> kp*

*kp* <sup>−</sup> *<sup>a</sup>*<sup>2</sup> *<sup>p</sup> kp*

<sup>2</sup> sin(*ϕ*−*ϕ*0) =

<sup>2</sup> sin(*ϕ*−*ϕ*0) cos *ϕ* =

<sup>2</sup> sin(*ϕ*−*ϕ*0) sin *ϕ* =

where

and

<sup>−</sup>*iα*<sup>1</sup> sin *<sup>ϕ</sup>*+*iα*<sup>2</sup> cos *<sup>ϕ</sup>* (38)

<sup>−</sup>*iα*<sup>1</sup> sin *<sup>ϕ</sup>*+*iα*<sup>2</sup> cos *<sup>ϕ</sup>* cos *ϕ* (39)

<sup>−</sup>*iα*<sup>1</sup> sin *<sup>ϕ</sup>*+*iα*<sup>2</sup> cos *<sup>ϕ</sup>* sin *ϕ*. (40)

∞ ∑ *s*=−∞

*Bse*

∞ ∑ *s*=−∞

∞ ∑ *s*=−∞

*i*(*s*−1)*ϕ*<sup>0</sup>  *B*1*se*

*B*2*se*

*isϕ*<sup>0</sup> (46)

, (41)

. (42)

<sup>−</sup>*is<sup>ϕ</sup>* (43)

<sup>−</sup>*is<sup>ϕ</sup>* (44)

<sup>−</sup>*isϕ*. (45)

(47)

$$\cos \varphi\_0 = \frac{\varkappa\_1}{z}; \quad \sin \varphi\_0 = \frac{\varkappa\_2}{z} \tag{50}$$

Functions *Bs*, *B*1*s*, *B*2*<sup>s</sup>* are related to one another as follows:

$$
\alpha\_1 B\_{1s} + \alpha\_2 B\_{2s} = sB\_{s\prime} \tag{51}
$$

which follows from the well-known relation for Bessel functions:

$$J\_{s-1}(z) + J\_{s+1}(z) = \frac{2s}{z} J\_s(z) \tag{52}$$

The matrix element (37) can be written in the form [12]

$$S\_{fi} = \frac{1}{\sqrt{2\omega' 2\eta\_0 2q\_0'}} \sum\_{s} M\_{fi}^{(s)}(2\pi)^4 \text{i}\delta^{(4)}(sk + q - q' - k'),\tag{53}$$

where the *δ*-function involves the law of conservation in the form

$$k\mathbf{k} + \mathbf{q} = \mathbf{q}' + \mathbf{k}',\tag{54}$$

where, respecting eq. (24),

$$q^{\mu} = p^{\mu} - \frac{e^2}{2(kp)} \overline{A^2} k^{\mu}. \tag{55}$$

Using the last equation, we can introduce the so-called effective mass of electron immersed in the periodic wave potential as follows:

$$q^2 = m\_\*^2; \quad m\_\* = m\sqrt{\left(1 - \frac{\varepsilon^2}{m^2}\overline{A^2}\right)}\tag{56}$$

Formula (56) represents the mass renormalization of an electron mass in the field *A*. In other words, the mass renormalization is defined by the equation

$$
\delta m\_{\text{phys}} = m\_{\text{bare}} + \delta m \tag{57}
$$

where *δm* follows from eq. (56). The quantity *m*phys is the physical mass that an experimenter would measure if the particle were subject to Newton's law **F** = *m*phys**a**. In case of the periodic field of laser, the quantity *δm* has the finite value. The renormalization is not introduced here "by hands," but it follows immediately from the formulation of the problem of electron in the wave field.

We can write

$$q^2 = q'^2 = m^2(1 + \tilde{\zeta}^2) \equiv m\_{\ast \prime}^2 \tag{58}$$

where for plane wave (35) with relations (36)

$$
\zeta = \frac{e}{m}\sqrt{-a^2}.\tag{59}
$$

It may be easy to see that eq. (58) has very simple limit for *m* = 0 which is the graphenic case. Or, in other words, with the help of eq. (59) *m*<sup>2</sup> <sup>∗</sup>(*<sup>m</sup>* <sup>=</sup> <sup>0</sup>) = <sup>−</sup>*e*2*a*2.

According to [12], the matrix element in (53) is of the form

$$\begin{split} M\_{fi}^{(s)} &= -e\sqrt{4\pi}\overline{u}(p') \left\{ \left( (\gamma e') - e^2 a^2 \frac{(ke')}{2(kp)} \frac{(\gamma k)}{(kp')} \right) B\_s + \\ & e \left( \frac{(\gamma a\_1)(\gamma k)(\gamma e')}{2(kp')} + \frac{(\gamma e')(\gamma k)(\gamma a\_1)}{2(kp)} \right) B\_{1s} + \\ e \left( \frac{(\gamma a\_2)(\gamma k)(\gamma e')}{2(kp')} + \frac{(\gamma e')(\gamma k)(\gamma a\_2)}{2(kp)} \right) B\_{2s} \right\} u(p) \end{split} \tag{60}$$

It is possible to show that the total probability of the emission of photons from unit volume in unit time is [12]

$$W = \frac{e^2 m^2}{4q\_0} \sum\_{s=1}^{\infty} \int\_0^{u\_s} \frac{du}{(1+u)^2} \quad \times$$

$$\left\{-4f\_s^2(z) + \xi^2 \left(2 + \frac{u^2}{1+u}\right) \left(f\_{s+1}^2(z) + f\_{s-1}^2(z) - 2f\_s^2(z)\right)\right\},\tag{61}$$

where

$$u = \frac{(kk')}{(kp')}, \quad u\_s = \frac{2s(kp)}{m\_s^2}, \quad z = 2sm^2\frac{\tilde{\xi}}{\sqrt{1+\tilde{\xi}^2}}\sqrt{\frac{u}{u\_s}\left(1-\frac{u}{u\_s}\right)}.\tag{62}$$

Variables *α*1,2 are to be expressed in terms of variables *u* and *us* from eq. (62).

When *ξ* 1 (the condition for perturbation theory to be valid), the integrand (61) can be expanded in powers of *ξ*. For the first term of the sum *W*1, we get

$$W\_1 \approx \frac{e^2 m^2}{4p\_0} \xi^2 \int\_0^{u\_1} \left[ 2 + \frac{u^2}{1+u} - 4 \frac{u}{u\_1} \left( 1 - \frac{u}{u\_1} \right) \right] \frac{1}{(1+u)^2} du = $$

$$\frac{e^2 m^2}{4p\_0} \xi^2 \left[ \left( 1 - \frac{4}{u\_1} - \frac{8}{u\_1^2} \right) \ln \left( 1 + u\_1 \right) + \frac{1}{2} + \frac{8}{u\_1} - \frac{1}{2(1+u\_1)^2} \right] \tag{63}$$

with *<sup>u</sup>*<sup>1</sup> <sup>≈</sup> <sup>2</sup>(*kp*)/*m*2. It is possible to determine the second and the next harmonics as an analogy with the Berestetzkii approach; however, the aim of this article was only to illustrate the influence of the dielectric medium on the Compton effect.

Let us consider eq. (54) in the form

Formula (56) represents the mass renormalization of an electron mass in the field *A*. In other

where *δm* follows from eq. (56). The quantity *m*phys is the physical mass that an experimenter would measure if the particle were subject to Newton's law **F** = *m*phys**a**. In case of the periodic field of laser, the quantity *δm* has the finite value. The renormalization is not introduced here "by hands," but it follows immediately from the formulation of the problem

*<sup>q</sup>*<sup>2</sup> <sup>=</sup> *<sup>q</sup>*<sup>2</sup> <sup>=</sup> *<sup>m</sup>*2(<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2) <sup>≡</sup> *<sup>m</sup>*<sup>2</sup>

It may be easy to see that eq. (58) has very simple limit for *m* = 0 which is the graphenic

) <sup>2</sup>(*kp*) <sup>+</sup> (*γ<sup>e</sup>*

It is possible to show that the total probability of the emission of photons from unit volume

*<sup>W</sup>* <sup>=</sup> *<sup>e</sup>*2*m*<sup>2</sup> 4*q*<sup>0</sup>

) <sup>2</sup>(*kp*) <sup>+</sup> (*γ<sup>e</sup>*

*<sup>ξ</sup>* <sup>=</sup> *<sup>e</sup> m* 

*m*phys = *m*bare + *δm* (57)

<sup>∗</sup>, (58)

*u*(*p*) (60)

, (61)

−*a*2. (59)

<sup>∗</sup>(*<sup>m</sup>* <sup>=</sup> <sup>0</sup>) = <sup>−</sup>*e*2*a*2.

)(*γk*)(*γa*1) 2(*kp*)

> *B*2*<sup>s</sup>*

(*γk*) (*kp*)  *Bs*+

 *B*1*<sup>s</sup>* +

*du* (<sup>1</sup> + *<sup>u</sup>*)<sup>2</sup> ×

> 2 *<sup>s</sup>* (*z*)

<sup>2</sup>*a*<sup>2</sup> (*ke* ) 2(*kp*)

)(*γk*)(*γa*2) 2(*kp*)

> ∞ ∑ *s*=1

> > 2

 *us* 0

*<sup>s</sup>*−1(*z*) <sup>−</sup> <sup>2</sup>*<sup>J</sup>*

words, the mass renormalization is defined by the equation

of electron in the wave field.

122 Graphene - New Trends and Developments

where for plane wave (35) with relations (36)

case. Or, in other words, with the help of eq. (59) *m*<sup>2</sup>

*M*(*s*) *f i* = −*e*

in unit time is [12]

where

 −4*J* 2 *<sup>s</sup>* (*z*) + *<sup>ξ</sup>*<sup>2</sup>

According to [12], the matrix element in (53) is of the form

√ 4*πu*¯(*p* ) (*γe* ) − *e*

(*γa*1)(*γk*)(*γe*

(*γa*2)(*γk*)(*γe*

 2 +

*u*2 1 + *u*  *J* 2 *<sup>s</sup>*+1(*z*) + *J*

*e*

*e*

We can write

$$
\mathbf{s}k + \mathbf{q} - \mathbf{k}' = \mathbf{q}'.\tag{64}
$$

Equation (64) has physical meaning for *s* = 1, 2, . . . *N*, *s*, *N* being positive integers. *s* = 1 means the conservation of energy momentum of the one-photon Compton process and *s* = 2 of the two-photon Compton process and *s* = *N* means the multiphoton interaction with *N* photons of laser beam with an electron. The multiphoton interaction is nonlinear and differs from the situation where electron scatters twice or more as it traverses the laser focus.

By analogy, the original Einstein photoelectric equation must be replaced by the more general multiphoton photoelectric equation in the form

$$
\hbar s \hbar \omega = \frac{1}{2} m v^2 + E\_{i\nu} \tag{65}
$$

where *Ei* is the binding energy of the outermost electron in the atomic system. It means that the ionization effect occurs also in the case that ¯*hω < Ei* in case that the number of participating photons is *s > Ei*/¯*hω*. We will not solve furthermore this specific problem.

We introduce the scattering angle *θ* between **k** and **k** . In other words, The scattering angle *θ* is measured with respect to the incident photon direction. Then, with |**k**| = *nω* and |**k** | = *nω* , where *n* is index of refraction of the dielectric, we get from the squared eq. (64) in the rest system of electron, where *q* = (*m*∗, 0), the following equation:

$$s\frac{1}{\omega'} - \frac{1}{\omega} = \frac{s}{m\_\*} (1 - n^2 \cos \theta) \,\prime \tag{66}$$

which is a modification of the original equation for the Compton process

$$\frac{1}{\omega'} - \frac{1}{\omega} = \frac{1}{m}(1 - \cos\theta). \tag{67}$$

Using relation *m*<sup>2</sup> <sup>∗</sup>(*<sup>m</sup>* <sup>=</sup> <sup>0</sup>) = <sup>−</sup>*e*2*a*2, we get eq. (66) for the situation of the Compton effect in the graphene sheet:

$$s\frac{1}{\omega'} - \frac{1}{\omega} = \frac{s}{m\_\*(m=0)}(1 - n^2 \cos \theta) = \frac{s}{e\sqrt{-a^2}}(1 - n^2 \cos \theta),\tag{68}$$

So, we see that the last Compton formula differs from the original one only by the existence of the renormalized mass and the occurrence of index of refraction.

We know that the original Compton formula can be written in the form suitable for the experimental verification, namely:

$$
\Delta\lambda = 4\pi \frac{\hbar}{mc} \sin^2 \frac{\theta}{2} \tag{69}
$$

which was used by Compton for the verification of the quantum nature of light [1]. The limiting case with *m* → 0 has the appropriate angle limit *θ* → 0.

If we consider the Compton process in dielectric, then the last formula goes to the following form:

$$
\Delta\lambda = 2\pi \frac{\hbar}{mc} (1 - n^2 \cos\theta). \tag{70}
$$

It is evident that relation *λ* − *λ* ≥ 0 follows from eq. (1). However, if we put

$$1 - n^2 \cos \theta \le 0,\tag{71}$$

or equivalently

$$\frac{1}{n^2} \le \cos \theta \le 1,\tag{72}$$

then we see that for some angles determined by eq. (72), the relation *λ* − *λ* ≤ 0 follows. This surprising result is the anomalous Compton effect which is caused by the index of refraction of the medium. To our knowledge, it was not published in the optical or particle journals.

The limiting case with *<sup>m</sup>* <sup>→</sup> 0 has the appropriate angle limit *<sup>n</sup>*<sup>2</sup> cos *<sup>θ</sup>* <sup>→</sup> 1. It means that there is of the angle shift in this case in comparison with the Compton effect in vacuum.

The equation *sk* + *q* = *q* + *k* is the symbolic expression of the nonlinear Compton effect in which several photons are absorbed at a single point, but only a single high-energy photon is emitted. The second process where electron scatters twice or more as it traverses the laser focus is not considered here. The nonlinear Compton process was experimentally confirmed, for instance, by Bulla et al. [15].

The formula (66) can be also expressed in terms of *λ* as follows:

$$
\lambda s \lambda' - \lambda = \frac{2\pi s}{m\_\*} (1 - n^2 \cos \theta) \tag{73}
$$

where we have put ¯*h* = *c* = 1. In the case of the graphene two-dimensional carbon sheet with zero mass of the pseudoelectron, we replace the renormalized mass by *m*∗(*m* = 0) = *e* √−*a*2, which is the new renormalized mass in graphene.

Formula (73) can be used for the verification of the Compton effect in a dielectric medium, and on the other hand, the index refraction follows from it in the following form:

$$m^2 = \frac{1}{\cos \theta} \left[ 1 - \frac{m\_\*}{2\pi s} (s\lambda' - \lambda) \right]. \tag{74}$$

It means, if we know the *θ*, *λ*, *λ* ,*s*, *m*∗, we are able to determine the index of refraction of some dielectric medium from the Compton effect. To our knowledge, this method was not published in the optical journals. In the graphene case, we write as usual *m*∗(*m* = 0) = *e* √−*a*<sup>2</sup> in the last formula.
