0. Recap: Widerøe and the cyclotron
In the previous lecture we learned how the Widerøe linear accelerator uses a sequence of RF gaps and drift tubes to overcome the limitations of simple DC acceleration. The same principle appears in the cyclotron: the particle are kept on circular trajectories by a magnetic dipole field. Every half turn they encounter the accelerating gap and gain energy repeatedly from a single RF source.
1. Alternating voltage or electromagnetic wave – what’s the difference anyway?
When discussing RF acceleration, we often speak of an “alternating voltage” applied to an accelerating gap. However, once the drive frequency becomes high, this picture becomes incomplete. An RF cavity is not merely a place where we impose a voltage — it is also an antenna, capable of emitting electromagnetic radiation.
Recall from Lecture 1 that a single accelerating gap with length \( d \approx \lambda/2 \) is appropriate for a particle that moves close to the speed of light. In such a configuration, the particle experiences an accelerating field over approximately half an RF cycle, which already ties the geometry of the accelerator to the wavelength of the driving field.
But what happens when we drive such a cavity with an AC voltage at frequency \( f \)? Any time-varying electric field inside the cavity generates time-varying charge and current distributions on the cavity walls. These, in turn, must radiate electromagnetic waves. Thus a cavity has two inseparable roles:
- Field shaping: it rectifies the RF field inside the gap.
- Radiation: it emits electromagnetic waves into free space, because the driven current distribution behaves as an antenna.
As we increase the frequency,
- the drift tubes and gaps become shorter (good for compact accelerators), but
- the cavity radiates more power into free space (because antenna efficiency improves when the structure approaches a significant fraction of a wavelength).
In other words, when we compactify an RF accelerator by raising the frequency,
we naturally transition from
“a voltage across electrodes” → “a high-frequency electromagnetic radiator”.
At low frequencies (kHz–MHz), cavities behave like classical voltage resonators. At microwave frequencies (GHz), they increasingly behave like antennas. At optical frequencies (hundreds of THz), they essentially are optical resonators and radiators.
Therefore, while higher frequency is beneficial for compactness, it also means the fields are no longer well confined — they tend to propagate as free electromagnetic waves.
This leads directly to the central question of this lecture: Why can’t a freely propagating electromagnetic wave accelerate a free electron?
2. Why free-space electromagnetic waves cannot accelerate particles
Consider an electron in a linearly polarized plane wave. In the non-relativistic limit,
\[ m\frac{dv}{dt} = qE_0\cos(\omega t) \quad \Rightarrow \quad v(t) = \frac{qE_0}{m\omega}\sin(\omega t). \]
The instantaneous energy transfer rate is
\[ \frac{dE_{\rm kin}}{dt} = \mathbf{v}\cdot\mathbf{F} \propto \sin(\omega t)\cos(\omega t), \]
which averages to zero over one cycle. Another way to write this is
\[ \Delta E_{\rm kin} \propto \Delta(v^2) \propto v\,\Delta v, \qquad v = v_{\rm max}\sin(\omega t), \] so that \[ \langle v\,\Delta v\rangle = 0. \]
This is a simple argument behind the Lawson–Woodward theorem: a free electron in free space cannot gain net energy from a plane electromagnetic wave.
Intuitively, this is the same phenomenon we saw in Lecture 1: a capacitor gap that is “too wide” gives zero net acceleration. A plane wave is simply the limit of an infinitely wide gap.
3. Rectifying the oscillating field
If an oscillating field is to accelerate particles, it must be rectified: the particle must only “see” the accelerating part of the cycle. Several strategies exist.
(a) Widerøe drift tubes
Drift tubes shield the particle during the wrong RF phase. Gap lengths obey \[ d_{\rm drift} \approx \frac{v}{2f_0}. \]
(b) Alvarez drift-tube linac (DTL)
A refinement by Luis Alvarez (1946): all drift tubes sit in a single resonant cavity. The TM mode enforces the correct axial field, enabling higher frequencies and much larger gradients.
(c) Microwave structures
Traveling-wave structures and disk-loaded waveguides phase-match the RF field to the particle velocity. These form the basis of modern linacs.
(d) THz and optical rectification
Recent work pushes the concept to extremely high frequencies: THz accelerators (e.g. at DESY) and dielectric laser accelerators (Hommelhoff; SLAC). They are, conceptually, optical analogues of the Widerøe idea.
However, all these “conventional” structures share a limitation: materials fail at very high fields. Lasers can reach field strengths exceeding atomic bonding forces, far beyond what metals or dielectrics can handle.
This motivates a completely different medium: plasmas as rectifiers. They cannot be damaged by strong fields because they are already fully ionized. This will be the basis of laser–plasma acceleration.
4. Why lasers? The role of peak power
Lasers are attractive not only because they operate at short wavelengths (and thus allow small structures), but also because they provide enormous peak power.
- Modern CPA lasers reach > 1 PW (e.g. ATLAS laser at CALA, LMU Munich).
- State-of-the-art microwave sources reach ≈ 100 GW. (For more on pulsed power see https://cds.cern.ch/record/261732/files/p913.pdf )
Why is power relevant for acceleration, not just the field? A very simple argument is
\[ \frac{dE_{\rm kin}}{dt} = \frac{dW}{dt} = P. \]
A more detailed derivation begins with \[ m\frac{dv}{dt} = qE_0\cos(\omega_0 t) \quad\Rightarrow\quad v(t) = \frac{qE_0}{m\omega_0}\sin(\omega_0 t), \] so the maximum velocity in half a cycle is \[ v_{\max} = \frac{qE_0}{m\omega_0}. \]
The oscillatory kinetic energy is \[ E_k = \tfrac12 m v_{\max}^2 = \frac{q^2E_0^2}{2m\omega_0^2}. \]
Using \[ I_0 = \frac{\varepsilon_0 c}{2}E_0^2, \] we find \[ E_k \propto I_0\lambda^2 \propto P. \] Thus the peak power directly limits the maximum energy extractable from the oscillatory motion.
5. Questions to think about
Consider a CPA laser system such as the Advanced Titanium:Sapphire Laser (ATLAS) at CALA, with
- peak power ≈ 1 PW,
- wavelength ≈ 800 nm,
- focus diameter ≈ 5 µm.
Questions for self-study:
- What is the peak intensity in such a focus?
- What temperature does this correspond to in thermal equilibrium?
- What is the corresponding electric field amplitude?
- What is the maximum oscillatory kinetic energy an electron can gain?
- Is the non-relativistic approximation still justified?
These questions will guide us toward understanding relativistic laser– matter interaction and the onset of laser–plasma acceleration.