the giant force of ultrashort-pulsed light
Electrons are bound to atoms or molecules with an extraordinarily strong force. Its strength can be estimated from the binding energy of the electrons (several electronvolts for outer (valence) electrons) and the sub-nanometer range of the atomic binding force as ~ binding energy / range of force , implying a binding field strength of several ten billion volt/meter! To affect the states of valence electrons in atoms or molecules, an external electric field of comparable strength must be applied. Such field strengths have been out of reach for any man-made technology until recently. With the advent of ultrashort laser pulses this unsatisfactory state of affairs changed radically. Indeed, hypershort flashes of laser light lasting merely a few field oscillation cycles – when focused with a curved mirror – confine energy in all three dimensions to a few micrometers. Thanks to this confinement, even pulses of moderate (several microjoules) energy are able to produce electric field strengths estimated above, see box. The electric field of intense few-cycle light, which is now available from compact sources
, is strong enough to steer the motion of valence electrons in atoms, molecules and solids.
, is strong enough to steer the motion of valence electrons in atoms, molecules and solids.The electric force of a light wave is directed parallel to the field indicated by the arrow (but opposite in direction because of the negative charge of the electron) and hence accelerates electrons in a direction perpendicular to the propagation direction of the laser pulse (from right to left). The magnetic force acting upon the transversally moving electrons acts in the forward direction, but is much weaker than the electric force (and hence negligible) unless the electrons are accelerated to a velocity comparable to the speed of light, by the electric force. This occurs, if the electric field becomes as strong as several trillion volts/meter or higher. This can be achieved in tightly-focused few-cycle light with pulse energies of tens of millijoules or higher, which are now becoming available.
.As a consequence, few-cycle light is able to ionize atoms with substantial probability within a single wave cycle, a key condition for the operation of current sources of isolated attosecond pulses , and accelerate electrons close to the speed of light in a reproducible fashion.
, and accelerate electrons close to the speed of light in a reproducible fashion.
The intensity of a light wave of electric field amplitude E_0 is given by I=E_0^2/2\,Z_0, where Z_0 = 377 {\rm V}/{\rm A} is a constant referred to as the “impedance of vacuum”. If the electric field is given in units of volt/cm ({\rm V}/{\rm cm}), I \approx 0.0013\,E_0^2 in units of {\rm watt}/{\rm cm}^2 ({\rm W}/{\rm cm}^2). Ionization and steering valence electrons in atoms and molecules requires an electric field strength comparable to the atomic binding force, i.e. E_{0,\rm ionizing} \ge \mbox{several} \times 10^8\:\mbox{V/cm}, i.e. intensities of I_{\rm ionizing} \ge 10^{14}\:\mbox{W}/{\rm cm}^2. A few-cycle laser pulse of duration of T_p = 5\,\mbox{fs} and energy of W_p=5\,{\rm \mu J}, if focused to a cross section of A_{\rm beam} = 1000\,{\rm \mu m}^2 (i.e. to a beam diameter of about 36 micrometers), yields this peak intensity:
I_{\rm peak} \approx \frac{1}{A}_{\rm beam}\,\frac{W_p}{T_p} = \frac{5\times10^{-6}\,{\rm J}}{10^{-5}{\rm cm}^2 \times 5 \times 10^{-15}\,{\rm s}} = 10^{14}\,\frac{\rm W}{{\rm cm}^2}
If we wish to accelerate electrons close to the speed of light, the electric force F_e of the light wave must be able to do a work within a half oscillation period of the light wave (as long as the force is directed in the same direction) on the electron that is comparable to its rest energy of W_0 \approx 0.5 {\rm} million e.V. Hence, for a temporally varying force, we have.
(F_{e,\rm max/2})\times (c/2) \times (T_{\rm osc}/2) = F_{e, \rm max} \, \lambda / 8 \ge W_0
For a light wave with a wavelength of \lambda \approx 0.8\,{\rm \mu m} this crude estimate yields an electric field amplitude of:E_{0,\rm relativistic} \approx 5 \times 10^{10}\:{\rm V/cm},
corresponding to an intensity of
I_{\rm relativistic} \approx 3 \times 10^{18}\:{\rm W}/{\rm cm}^2.In a freely propagating light wave, the magnetic field is connected to the electric field by the speed of light:
B=E/c.
As a consequence, the magnetic force which is directed perpendicularly to the velocity of the electron and the magnetic field direction and has the strength F_m=e\,v\,B , becomes, for an electron moving at V \approx c comparable to the electric field
F_m = e\,v\,B \approx e\, c\, B = e\ E = F_e
in strength, but directed in the forward direction, bending the trajectory of the relativistic electron in the direction of laser pulse propagation.These intensities are now routinely attainable with conventional high-power femtosecond lasers, producing powerful multi-cycle laser pulses with durations of tens to hundreds of femtoseconds. However, the large number of wave cycles have a detrimental effect on the both controlling bound electron dynamics as well as for laser-driven electron acceleration. In the former case the system is usually ionized by the long leading edge of the pulse, before it is subjected to the most intense field oscillation cycles, preventing, among others, efficient attosecond pulse generation. In the latter case, few-cycle pulses are ideal for laser-plasma accelerators.
. In the latter case, few-cycle pulses are ideal for laser-plasma accelerators.