Interaction of Ultrashort Electromagnetic Pulses with Matter: Description in the Framework of Perturbation Theory

Abstract

Considerable advances have been made in the generation of ultrashort electromagnetic field pulses of controlled shape over a wide spectral range [1]. In the infrared, visible, and far-ultraviolet spectral regions, pulses have been produced with widths equal to the period of the electromagnetic field oscillation at the carrier frequency (single-cycle pulses).

Appendix III Dynamic Form Factor of Plasma Particles

The dynamic form factor (DFF) defines the probability of electromagnetic interactions with participation of plasma particles, during which the subsystem of plasma electrons or ions absorbs the energy–momentum excess. Such processes are exemplified by radiation scattering in plasma, bremsstrahlung and polarization bremsstrahlung on plasma particles including the stimulated bremsstrahlung effect, and a number of other phenomena.

The determination of the DFF of a specified plasma component has the form

$$ S\left( {\omega ,\,{\mathbf{k}}\,} \right) \,=\, \frac{1}{2\,\pi }\,\int\limits_{ - \infty }^{\infty } {dt\,e^{i\,\,\omega t} \,\left\langle {\hat{n}\left( {{\mathbf{k}},t} \right)\,\hat{n}\left( { - {\mathbf{k}}} \right)} \right\rangle }, $$

(A.1)

where $\hat{n}\left( {\mathbf{k}} \right),\;\hat{n}\left( {{\mathbf{k}},\,t} \right)$ are spatial Fourier transforms of the operators representing the concentration of plasma particles of a specified type in the Schrödinger and Heisenberg pictures, and the angle brackets include both quantum–mechanical and statistical averages.

It will be recalled that the Heisenberg representation of quantum–mechanical operators takes into account their time dependence, in contrast to the Schrödinger representation, in which the whole time dependence is transferred to the wave function of the system. The relationship between these representations for an arbitrary operator $\hat{Q}$ is

$$ \hat{Q}\left( t \right) \,=\, \exp \left( {{{i\,\hat{\rm H}\,t} \mathord{\left/ {\vphantom {{i\,\hat{\rm H}\,t} \hbar }} \right. \kern-0pt} \hbar }} \right)\,\hat{Q}\,\exp \left( {{{ - i\,\hat{\rm H}\,t} \mathord{\left/ {\vphantom {{ - i\,\hat{\rm H}\,t} \hbar }} \right. \kern-0pt} \hbar }} \right), $$

where $\hat{\rm H}$ is the Hamiltonian of the quantum–mechanical system. In this appendix, however, the quantum–mechanical formalism will not be used, and the quantum description is given only for completeness. Equation (A.1) can be obtained from the formula

$$ S\left( {\omega ,\,{\mathbf{k}}\,} \right) \,=\, \sum\limits_{f,i} {w\left( i \right)\,\delta \left( {\omega \,+\, \omega_{fi} } \right)\,\left| {n_{fi} \left( {\mathbf{k}} \right)} \right|^{2} } , $$

(A.2)

averaging over initial states $\left| i \right\rangle$ and summing over final states $\left| f \right\rangle$ of the plasma particles (w(i) is the probability of a plasma particle being in the ith state). As usual the delta function in (A.2) reflects energy conservation.

Depending on the type of plasma particles, the DFF can be electronic, ionic, or mixed. For the mixed DFF, the product of the density operators for electrons and ions appears in the determination of (A.1).

Physically, the DFF defines the probability of plasma absorption of the four-dimensional wave vector k = (ω, k) in terms of the action of an external disturbance on a specified plasma component. When the charge distribution in the plasma is uniform, this probability is equal to zero, since then the Fourier transform of the density of charged particles reduces to the delta function $n\left( {\mathbf{k}} \right) \to n\,\delta \left( {\mathbf{k}} \right).$ Thus the DFF is connected with charge fluctuations in the plasma.

In fact, the dynamic form factor reflects the dynamics of plasma particles interacting with each other through long-range Coulomb forces. Interactions are then taken into account within the ensemble of one type of particles and also between electrons and ions.

For an uniform plasma, it is convenient to introduce the DFF of the unit volume (the normalized DFF) by the formula

$$ \tilde{S}\left( {\omega ,\,{\mathbf{k}}\,} \right) \,=\, \frac{{S\left( {\omega ,\,{\mathbf{k}}\,} \right)}}{V}, $$

(A.3)

where V is the volume of the plasma. This equation follows from the fact that, for an uniform medium, the pair correlation function of the concentration depends only on the relative distance between spatial points:

$$ Kn\left( {{\mathbf{r}},{\mathbf{r^{\prime}}},t} \right) \equiv \left\langle {\hat{n}\left( {{\mathbf{r}},t} \right)\,\hat{n}\left( {{\mathbf{r^{\prime}}},0} \right)} \right\rangle \,=\, Kn\left( {{\mathbf{r}} \,-\, {\mathbf{r^{\prime}}},t} \right). $$

To calculate the normalized DFF, it is convenient to use the fluctuation–dissipation theorem connecting the DFF of a plasma component with the function describing the plasma response to the external electromagnetic disturbance [8]. This theorem for the electron DFF is expressed by

$$ \tilde{S}_{e} \left( {\omega ,\,{\mathbf{k}}} \right)\; = \;\frac{\hbar }{{\pi \,e^{2} }}\frac{{\;\text{Im} \left\{ {F_{ee} \left( {\omega ,\,{\mathbf{k}}} \right)} \right\}}}{{\,\left[ {\exp \left( { - \hbar {\omega \mathord{\left/ {\vphantom {\omega T}} \right. \kern-0pt} T}} \right) - 1} \right]}} $$

(A.4)

where $F_{ee} \left( {\omega ,\,{\mathbf{k}}} \right)$ is the linear function describing the electron component response to the fictitious external potential acting only on plasma electrons, and T is the temperature of the plasma in energy units. The imaginary part of the response function appearing in (A.4) describes energy dissipation in the plasma, whence the name for the theorem.

With reference to [8], we introduce a second linear function describing the response to the external potential $F_{ei} \left( {\omega ,\,{\mathbf{k}}} \right),$ i.e., describing the response of the electron component of the plasma to the action of the fictitious external potential acting only on plasma ions. Here for convenience we use the Coulomb gauge of the electromagnetic field, in which the divergence of the vector potential is equal to zero (divA = 0) and the charge density is related only to the scalar potential of the electromagnetic field φ via the Poisson equation. So the external potential φ _ext (k) acts on the plasma, where k = (ω, k) is the four-dimensional wave vector. Then the density of the electron charge induced in the plasma is expressed in terms of the above response functions as follows:

$$ \left\langle {\hat{\rho }_{e} \left( k \right)} \right\rangle \,=\, \left[ {F_{ee} \left( k \right) \,+\, F_{ei} \left( k \right)} \right]\,\varphi_{ext} \left( k \right). $$

(A.5)

$\left\langle {\hat{\rho }_{j} \left( k \right)} \right\rangle \,=\, e_{j} \,\left\langle {\hat{n}_{j} \left( k \right)} \right\rangle$ is the charge density of the jth type of plasma particles. Equation (A.5) indicates that the electron charge density arises in the plasma due to direct action of the external potential on plasma electrons [the first summand in the square brackets of (A.5)] and also as a result of the action of the external potential on plasma ions that are bound to electrons by Coulomb forces. If interaction between particles of type i and type j is weak, one can use the technique described in [8] to express F _ij in terms of the characteristics of noninteracting particles. For this purpose the new response function α _j (k) is introduced—the response function for particles of type j to the total potential in the plasma. It takes into account the action on charged particles of the potential φ _ind (k) induced in the plasma due to redistribution of the charged particles under the action of the external potential. With the help of the function α _j (k), the induced charge density for the jth component can be expressed in terms of the total potential:

$$ \left\langle {\hat{\rho }_{j} \left( k \right)} \right\rangle \,=\, \alpha_{j} \left( k \right)\,\varphi_{tot} \left( k \right). $$

(A.6)

As the response function α _j (k) describes the action of the total potential on the plasma particles, the characteristics of noninteracting particles can be used to calculate it, since interaction between them is already taken into account in the total potential. This technique is widely used in plasma physics to describe screening and initiation of collective excitations. In the approach under consideration, corresponding to the random phase approximation [8], α _j  (k) can be expressed in terms of the function $Q_{j} \left( k \right)$ characterizing the noninteracting particles according to $\alpha_{j} \,=\, e_{j}^{2} \,Q_{j} ,$ where

$$ Q_{j} \left( k \right) \,=\, \int {\frac{{n_{j} \left( {{\mathbf{p}} \,+\, \hbar \,{\mathbf{k}}} \right) \,-\, n_{j} \left( {\mathbf{p}} \right)}}{{{\rm E}_{j} \left( {{\mathbf{p}} \,+\, \hbar \,{\mathbf{k}}} \right) \,-\, {\rm E}_{j} \left( {\mathbf{p}} \right) \,-\, \hbar \,\omega \,-\, i\,0}}} \,\,\frac{{2\,d{\mathbf{p}}}}{{\left( {2\,\pi \,\hbar } \right)^{3} }}. $$

(A.7)

Here n _j (p) is the dimensionless momentum distribution function of plasma particles of type j and ${\rm E}_{j} \left( {\mathbf{p}} \right) \,=\, {{p^{2} } \mathord{\left/ {\vphantom {{p^{2} } {2\,m_{j} }}} \right. \kern-0pt} {2\,m_{j} }}.$ Hereafter we need to know the imaginary part of the function $Q_{j} \left( k \right),$ which can be determined from (A.7) using the Sokhotsky formula. For the Maxwell velocity distribution of the electrons, we find

$$ \text{Im} \left\{ {Q_{j} \left( k \right)} \right\} \,=\, \pi \,\left( {e^{{ - {{\hbar \,\omega } \mathord{\left/ {\vphantom {{\hbar \,\omega } T}} \right. \kern-0pt} T}}} \,-\, 1} \right)\,n_{j} \,\frac{{\exp \left\{ { - {{\omega^{2} } \mathord{\left/ {\vphantom {{\omega^{2} } {2\,k^{2} \,{\text{v}}_{Tj}^{2} }}} \right. \kern-0pt} {2\,k^{2} \,{\text{v}}_{Tj}^{2} }}} \right\}}}{{\sqrt {2\,\pi } \,k\,{\text{v}}_{Tj} }}. $$

(A.8)

The functions introduced above to describe the response to the total potential are related to the longitudinal part of the dielectric permittivity by

$$ \varepsilon^{{\left( {l,j} \right)}} \left( k \right) \,=\, 1 \,-\, \frac{4\,\pi }{{{\mathbf{k}}^{2} }}\,\alpha_{j} \left( k \right). $$

(A.9)

We can now solve the original problem, i.e., we will find the function F _ee (ω, k) and express it in terms of the function describing the response to the total potential. For this purpose we introduce the fictitious external potential $\varphi_{ext}^{ * }$ acting only on electrons. Then according to the definition of F _ee (ω, k), we have

$$ \left\langle {\hat{\rho }_{e}^{ * } \left( k \right)} \right\rangle \,=\, F_{ee} \left( k \right)\,\varphi_{ext}^{ * } \left( k \right). $$

(A.10)

On the other hand, $\left\langle {\hat{\rho }_{e}^{ * } } \right\rangle$ can be expressed in terms of α _e :

$$ \left\langle {\hat{\rho }_{e}^{ * } \left( k \right)} \right\rangle \,=\, \alpha_{e} \left( k \right)\,\left[ {\varphi_{ext}^{ * } \left( k \right) \,+\, \varphi_{ind}^{ * } \left( k \right)} \right], $$

(A.11)

where $\varphi_{ind}^{ * }$ is the potential induced under the action of $\varphi_{ext}^{ * } ,$ determined in terms of the density of all plasma charges with the help of the Poisson equation:

$$ \varphi_{ind}^{ * } \left( k \right) \,=\, \frac{4\,\pi }{{{\mathbf{k}}^{2} }}\,\left[ {\left\langle {\hat{\rho }_{e}^{ * } \left( k \right)} \right\rangle \,+\, \left\langle {\hat{\rho }_{i}^{ * } \left( k \right)} \right\rangle } \right]. $$

(A.12)

Here

$$ \left\langle {\hat{\rho }_{i}^{ * } \left( k \right)} \right\rangle \,=\, \alpha_{i} \left( k \right)\,\varphi_{ind}^{ * } \left( k \right), $$

(A.13)

Since the potential $\varphi_{ext}^{ * }$ is assumed to act only on electrons. Solving the system of Eqs. (A.8–A.12), we find the following expression for F _ee :

$$ F_{ee} \left( k \right) \,=\, \frac{{\alpha_{e} \left( k \right)\,\left[ {1 \,-\, \left( {{{4\,\pi } \mathord{\left/ {\vphantom {{4\,\pi } {{\mathbf{k}}^{2} }}} \right. \kern-0pt} {{\mathbf{k}}^{2} }}} \right)\,\alpha_{i} \left( k \right)} \right]}}{{1 \,-\, \left( {{{4\,\pi } \mathord{\left/ {\vphantom {{4\,\pi } {{\mathbf{k}}^{2} }}} \right. \kern-0pt} {{\mathbf{k}}^{2} }}} \right)\;\left[ {\alpha_{e} \left( k \right) \,+\, \alpha_{i} \left( k \right)} \right]}}. $$

(A.14)

Substituting (A.13) into (A.4) and using (A.8) and (A.7), we obtain

$$ \tilde{S}_{e} \left( k \right) \,=\, \left| {\frac{{\varepsilon^{l\left( i \right)} \left( k \right)}}{{\varepsilon^{l} \left( k \right)}}} \right|^{2} \left| {\delta n_{e} \left( k \right)} \right|^{2} \,+\, z_{i}^{2} \left| {\frac{{1 \,-\, \varepsilon^{l\left( e \right)} \left( k \right)}}{{\varepsilon^{l} \left( k \right)}}} \right|^{2} \,\left| {\delta n_{i} \left( k \right)} \right|^{2} , $$

(A.15)

where

$$ \left| {\delta n_{e,i} \left( k \right)} \right|^{2} \,=\, \frac{{n_{e,i} }}{{\sqrt {2\pi } \,{\text{v}}_{Te} \,\left| {\mathbf{k}} \right|}}\,\exp \left( { - \frac{{\omega^{2} }}{{2\,{\mathbf{k}}^{2} \,{\text{v}}_{Te,i}^{ 2} }}} \right) $$

(A.16)

are the spatio-temporal Fourier transforms of the squared thermal fluctuations of the electron and ionic components of the plasma calculated for the four-dimensional wave vector $k \,=\, \left( {{\mathbf{k}},\,\omega } \right).$ z _i is the charge number of the plasma ions and it is implied that the quasi-neutrality condition is satisfied, so that n _e  = z _i  n _i .

The expression for the normalized ionic DFF is found in exactly the same way as the electron DFF. For this purpose, one must make the index replacement $e \rightleftarrows i$ and take into account the fact that, in the denominator of (A.4), the ion charge $e_{i} \,=\, z_{i} \,e$ now appears. Then we obtain:

$$ \tilde{S}_{i} \left( k \right) \,=\, \left| {\frac{{\varepsilon^{l\left( e \right)} \left( k \right)}}{{\varepsilon^{l} \left( k \right)}}} \right|^{2} \left| {\delta n_{i} \left( k \right)} \right|^{2} \,+\, z_{i}^{ - 2} \left| {\frac{{1 \,-\, \varepsilon^{l\left( i \right)} \left( k \right)}}{{\varepsilon^{l} \left( k \right)}}} \right|^{2} \,\left| {\delta n_{e} \left( k \right)} \right|^{2} . $$

(A.17)

The mixed normalized DFF is given by

$$ \tilde{S}_{ei} \left( k \right) \,=\, z_{i}^{ - 1} \left| {\frac{{1 \,-\, \varepsilon^{l\left( i \right)} \left( k \right)}}{{\varepsilon^{l} \left( k \right)}}} \right|^{2} \,\left| {\delta n_{e} \left( k \right)} \right|^{2} \,+\, z_{i} \left| {\frac{{1 \,-\, \varepsilon^{l\left( e \right)} \left( k \right)}}{{\varepsilon^{l} \left( k \right)}}} \right|^{2} \left| {\delta n_{i} \left( k \right)} \right|^{2} , $$

(A.18)

which follows from the fluctuation–dissipation theorem (A.4) (with the replacement $e^{2} \to e\,e_{i}$) and the formula for the linear response function $F_{ei}$ describing the initiation of an electron charge induced by the fictitious potential that acts only on ions. This formula has the form

$$ F_{ei} \left( k \right) \,=\, \frac{{\left( {{{4\,\pi } \mathord{\left/ {\vphantom {{4\,\pi } {{\mathbf{k}}^{2} }}} \right. \kern-0pt} {{\mathbf{k}}^{2} }}} \right)\,\alpha_{i} \left( k \right)\alpha_{e} \left( k \right)\,}}{{1 \,-\, \left( {{{4\,\pi } \mathord{\left/ {\vphantom {{4\,\pi } {{\mathbf{k}}^{2} }}} \right. \kern-0pt} {{\mathbf{k}}^{2} }}} \right)\;\left[ {\alpha_{e} \left( k \right) \,+\, \alpha_{i} \left( k \right)} \right]}}. $$

(A.19)

Equation (A.18) is obtained by analogous reasoning to the deduction of (A.13).

Let us explain the physical meaning of the expression (A.14) for the electron DFF. The first summand is connected with the deficiency of electron charge around the electron density fluctuation, caused by electron–electron repulsion. The second summand in this expression describes the electron charge screening the fluctuation of the ionic plasma component. It results from electron–ion attraction. By analogy, in the expression (A.16) for the ionic DFF, the second summand describes the ionic charge screening the electron density fluctuation, while the first summand describes the deficiency of ionic charge around the ionic fluctuation. Finally, in the formula (A.16) for the mixed DFF, the first summand describes the ionic charge screening the electron density fluctuation, and the second summand describes the electron charge screening the ionic density fluctuation.

Let us consider the explicit form of the electron DFF satisfying the inequations $k\,{\text{v}}_{Te} \gg \omega \gg k\,{\text{v}}_{Ti} ,\;\omega_{pi} .$ Then the low frequency approximation is valid for the longitudinal electron dielectric permittivity of the plasma, and the high-frequency approximation is valid for the ionic component. Using the expressions for the longitudinal part of the dielectric permittivity of the plasma and the formula (A.15), we find

$$ \tilde{S}_{e} \left( k \right) \simeq \left( {\frac{{k^{2} \,r_{De}^{2} }}{{1 \,+\, k^{2} \,r_{De}^{2} }}} \right)^{2} \left| {\delta n_{e} \left( k \right)} \right|^{2} \,+\, \frac{{z_{i}^{2} }}{{\left( {1 \,+\, k^{2} \,r_{De}^{2} } \right)^{2} }}\,\left| {\delta n_{i} \left( k \right)} \right|^{2} . $$

(A.20)

From this formula we see that, in the case of long-wavelength fluctuations, when $k^{2} \,r_{De}^{2} \, \ll \,1$ $\left( {k \,=\, {{2\,\pi } \mathord{\left/ {\vphantom {{2\,\pi } \lambda }} \right. \kern-0pt} \lambda }} \right),$ the first summand describing the deficiency of electron charge around the electron density fluctuation is small. The second summand connected with electron screening of ionic density fluctuations is large. Hence it follows that, in the long-wavelength limit, the transfer of energy–momentum to the plasma proceeds through the electron charge of the Debye sphere around a plasma ion which reacts in a coherent manner to the electric field. That is, interaction is of a collective nature. In the short-wavelength case $k^{2} \,r_{De}^{2} \, \gg \,1,$ the situation is opposite: electromagnetic interaction is realized through excitation of individual plasma electrons, under which the Debye sphere “falls apart” due to the strong spatial non-uniformity of the electric field.