1. Equation of motion in an electromagnetic field
We now focus on the motion of a free electron in a laser field. The starting point is the Lorentz force equation,
\[ \frac{d\mathbf{p}}{dt} = q\left(\mathbf{E} + \mathbf{v}\times\mathbf{B}\right), \]
where \(\mathbf{p} = \gamma m \mathbf{v}\) is the relativistic momentum and \(q=-e\) for an electron. The energy equation can be written as
\[ m c^2 \frac{d\gamma}{dt} = q\,\mathbf{v}\cdot\mathbf{E} = -e\,\mathbf{v}\cdot\mathbf{E}. \]
In a linearly polarized plane wave, propagating along the \(x\)-direction with vector potential \(\mathbf{A}(\xi)\) that depends only on the phase \(\xi = \omega_0 t - k_0 x\), one can solve these equations analytically (see e.g. Gibbon, Short Pulse Laser Interactions with Matter). Two important invariants of the motion are:
- \(\mathbf{p}_\perp - q\mathbf{A}_\perp/c = \text{const}\) (transverse canonical momentum),
- \(\gamma - \frac{p_x}{m c} = \text{const}\) (light-front invariant for a plane wave).
For an electron initially at rest, these invariants completely determine the trajectory and energy gain in a plane wave. As discussed earlier, a free electron cannot gain net energy from an infinite plane wave, but its instantaneous (ponderomotive) energy can be very large.
Interactive tool: electron in a plane-wave pulse
The following interactive simulation shows the motion of an electron in a plane-wave laser pulse. You can change the pulse duration, carrier frequency, peak amplitude, duration, etc and observe how the electron trajectory and energy evolve.
2. Temporally Gaussian laser pulse (still plane wave)
To move from a monochromatic plane wave to a realistic laser pulse, we first keep the spatial dependence as a plane wave but introduce a finite temporal envelope. A convenient model is a Gaussian pulse in time:
\[ \mathbf{E}(t,x) = \hat{\mathbf{y}}\, E_0\, \exp\!\left[-\frac{(t - x/c)^2}{\tau^2}\right] \cos\!\left(\omega_0 (t - x/c) + \varphi_{\text{CEP}}\right), \]
where
- \(E_0\) is the peak electric field amplitude,
- \(\tau\) is a measure of the pulse duration (e.g. related to FWHM),
- \(\omega_0\) is the central angular frequency,
- \(\varphi_{\text{CEP}}\) is the carrier–envelope phase,
- \(\hat{\mathbf{y}}\) is the polarization direction.
The corresponding magnetic field of the plane wave is \[ \mathbf{B}(t,x) = \hat{\mathbf{z}}\, \frac{E_0}{c}\, \exp\!\left[-\frac{(t - x/c)^2}{\tau^2}\right] \cos\!\left(\omega_0 (t - x/c) + \varphi_{\text{CEP}}\right), \] so that \(|\mathbf{B}| = |\mathbf{E}|/c\) and \(\mathbf{B} \perp \mathbf{E} \perp \mathbf{k}\).
This model already captures important effects of the finite duration for the instantaneous dynamics: the electron experiences a time-dependent ponderomotive potential and can reach large transient momenta and energies while the pulse is present. However, as long as the field remains a perfect plane wave with an envelope that depends only on \((t - x/c)\), the plane-wave invariants still guarantee that an electron which starts and ends outside the pulse cannot gain net energy or net momentum from the pulse.
3. Spatial focus: Gaussian beam and longitudinal fields
Real laser pulses are not only finite in time but also spatially focused. A standard model is the Gaussian beam in the paraxial approximation. For a beam propagating along \(x\), linearly polarized in \(y\)-direction, the complex field of a continuous wave can be written as
\[ E_y(r,x,t) = \Re\Bigg\{ E_0\, \frac{w_0}{w(x)} \exp\!\left[-\frac{r^2}{w^2(x)}\right] \exp\!\left[-i\big( \omega_0 t - k_0 x - \frac{k_0 r^2}{2R(x)} + \zeta(x) \big)\right] \Bigg\}, \]
with
- \(r^2 = y^2 + z^2\) (radial coordinate),
- \(w_0\) beam waist at focus,
- \(w(x) = w_0 \sqrt{1 + (x/x_R)^2}\) beam radius,
- \(x_R = \pi w_0^2/\lambda_0\) Rayleigh length,
- \(R(x) = x \big[1 + (x_R/x)^2\big]\) wavefront radius of curvature,
- \(\zeta(x) = \arctan(x/x_R)\) Gouy phase.
A temporally Gaussian focused pulse can be modeled by multiplying this spatial envelope with a temporal Gaussian:
\[ E_y(r,x,t) = \Re\Bigg\{ E_0\, \frac{w_0}{w(x)} \exp\!\left[-\frac{r^2}{w^2(x)}\right] \exp\!\left[-\frac{(t - x/c)^2}{\tau^2}\right] e^{-i\big( \omega_0 t - k_0 x - \frac{k_0 r^2}{2R(x)} + \zeta(x) \big)} \Bigg\}. \]
Important: a focused beam is not obtained by simply “multiplying a plane wave with a Gaussian in space”. Maxwell’s equations require that spatial variation of the transverse components induces longitudinal fields. In particular, there are non-zero components \(E_x\) and \(B_x\) of order \(1/(k_0 w_0)\), which ensure that \(\nabla\cdot\mathbf{E}=0\) and \(\nabla\times\mathbf{E}=-\partial_t\mathbf{B}\) are satisfied consistently.
These longitudinal fields have important consequences for the motion of an electron near the focus: they contribute to the net drift and to the effective ponderomotive force that pushes particles out of high-intensity regions.
Interactive tool: electron in a focused Gaussian pulse
The next interactive simulation shows a free electron in a focused, Gaussian laser pulse (first order paraxial). By varying focus size, pulse duration, and peak intensity you can observe how the electron is pushed out of the focal region by the combined effect of the transverse electric field and the longitudinal magnetic field. Cycle-average motion is indicated by the green lines.
4. Ponderomotive force in an inhomogeneous oscillating field
As an exercise, we derive the ponderomotive force in the non-relativistic limit for a charged particle in an oscillating electric field with spatially varying amplitude. We consider a particle with charge \(q = Z e\) and mass \(m = R m_e\), where \(Z\) is the charge number and \(R\) is the mass in units of the electron mass.
The key assumptions are:
- During one laser cycle, the particle displacement \(\Delta x\) is small compared to the scale on which the field amplitude varies: \(\Delta x \ll L_E\).
- The oscillation velocity is non-relativistic, \(v \ll c\).
We take the electric field to have the form \[ \mathbf{E}(\mathbf{r},t) = \hat{\mathbf{e}}\, E_0(\mathbf{r}) \cos(\omega_0 t), \] where \(\hat{\mathbf{e}}\) is a fixed polarization direction and \(E_0(\mathbf{r})\) is a slowly varying amplitude (e.g. the envelope of a focused beam). The equation of motion is
\[ m \frac{d\mathbf{v}}{dt} = Z e \left( \mathbf{E} + \mathbf{v} \times \mathbf{B} \right). \]
Because the field is spatially varying, Maxwell’s equations require an associated magnetic field. For our purposes, the relevant Maxwell equation is \[ \nabla\times\mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}, \] which allows us to express \(\mathbf{B}\) in terms of the spatial derivatives of \(\mathbf{E}\). We treat the particle as a test particle (no back-reaction on the field).
4.1 First-order motion
To lowest order, we neglect the spatial variation and the magnetic term. The first-order equation of motion is \[ m \frac{d\mathbf{v}_1}{dt} = Z e\, \hat{\mathbf{e}}\, E_0(\mathbf{r}_0)\cos(\omega_0 t), \] where \(\mathbf{r}_0\) is the initial position. Integrating, we obtain
\[ \mathbf{v}_1(t) = \frac{Z e}{m\omega_0}\, \hat{\mathbf{e}}\, E_0(\mathbf{r}_0)\, \sin(\omega_0 t), \] \[ \mathbf{x}_1(t) = -\frac{Z e}{m\omega_0^2}\, \hat{\mathbf{e}}\, E_0(\mathbf{r}_0)\, \cos(\omega_0 t), \]
so that the oscillation amplitude is \[ \Delta x \sim \frac{Z e E_0}{m \omega_0^2}. \]
The particle oscillates at the driving frequency \(\omega_0\) and remains approximately centered around \(\mathbf{r}_0\) on short time scales.
4.2 Second-order force and cycle averaging
To obtain the slow drift motion, we must include the spatial variation of \(E_0(\mathbf{r})\) and the associated magnetic field. Expanding the electric field around \(\mathbf{r}_0\), \[ E_0(\mathbf{r}) \approx E_0(\mathbf{r}_0) + (\mathbf{x}_1\cdot\nabla) E_0(\mathbf{r}_0) + \cdots, \] and inserting \(\mathbf{x}_1(t)\) into the force equation, we collect all terms that are second order in the oscillation amplitude. The second-order equation for the slow velocity \(\mathbf{v}_2\) has the schematic form
\[ m \frac{d\mathbf{v}_2}{dt} = Z e \left[ (\mathbf{x}_1\cdot\nabla)\mathbf{E} + \mathbf{v}_1 \times \mathbf{B}_1 \right], \]
where both terms are proportional to \(E_0\nabla E_0\). When we average over one oscillation cycle, all terms oscillating at frequency \(\omega_0\) vanish, and we are left with a secular (slow) force. For linear polarization, the result is
\[ \langle \mathbf{F}_p \rangle = -\,\frac{Z^2 e^2}{4 m \omega_0^2}\, \nabla E_0^2(\mathbf{r}_0), \]
which is the ponderomotive force. It always points towards regions of lower field amplitude, i.e. it pushes the particle out of high-intensity regions such as the focus of a laser beam.
4.3 Circular polarization
For a circularly polarized wave propagating along \(x\), we can write the electric field as \[ \mathbf{E}(t) = E_0(\mathbf{r}) \big[ \hat{\mathbf{y}}\cos(\omega_0 t) + \hat{\mathbf{z}}\sin(\omega_0 t) \big], \] with the corresponding magnetic field determined by \(\mathbf{B} = \mathbf{k}\times\mathbf{E}/\omega_0\). Repeating the perturbative calculation, one finds a similar second-order force:
\[ \langle \mathbf{F}_p \rangle_{\text{circ}} = -\,\frac{Z^2 e^2}{2 m \omega_0^2}\, \nabla E_0^2(\mathbf{r}_0), \]
i.e. the same structure as for linear polarization, but with a factor of 2. This is related to the fact that, for a given time-averaged intensity, the peak field amplitude in circular polarization is reduced by \(1/\sqrt{2}\) compared to linear polarization.
4.4 Ponderomotive potential and scaling with intensity
The cycle-averaged oscillatory kinetic energy of the particle in a linearly polarized field is \[ U_p = \left\langle \frac{1}{2} m v_1^2 \right\rangle = \frac{Z^2 e^2 E_0^2}{4 m \omega_0^2}. \] This quantity is called the ponderomotive potential because the ponderomotive force can be written compactly as \[ \langle \mathbf{F}_p \rangle = -\nabla U_p. \]
For circular polarization, the average kinetic energy is the same, but the relation between \(U_p\) and the intensity differs by a factor of 2. Using the plane-wave relation between intensity and field amplitude, \(I_0 = \tfrac{1}{2}\varepsilon_0 c E_0^2\), we can express \(U_p\) as
\[ U_p = \frac{Z^2 e^2 E_0^2}{4 m \omega_0^2} = \frac{Z^2 e^2}{2 m \varepsilon_0 c \omega_0^2} I_0 \propto \frac{Z^2}{m}\, I_0 \lambda_0^2. \]
It is often convenient to introduce the classical electron radius \(r_e = e^2/(4\pi\varepsilon_0 m_e c^2)\) and write
\[ U_p = \frac{2\pi Z^2 r_e m_e c}{\omega_0^2 m}\, I_0 = \left(\frac{Z^2}{R^2}\right) \left(\frac{I_0 \lambda_0^2}{2 \pi P_{\text{rel}}}\right) m c^2, \]
where \(R=m/m_e\) and \(P_{\text{rel}}\) is the (electron) relativistic power unit discussed in Lecture 4. This form makes it transparent at which combination of intensity and wavelength the ponderomotive energy becomes comparable to the rest energy of the particle.