- A freely propagating plane wave laser pulse is like an infinitely wide RF gap: the particle stays in phase with the wave — no net energy gain (Lawson–Woodward).
- Acceleration requires a converter structure that rectifies the wave: drift tubes (Widerøe), cavities (Alvarez DTL), disk-loaded waveguides — or, at optical frequencies, a dielectric grating.
- A completely different route is light-sail acceleration: instead of an external field acting on a few beam particles, photon momentum pushes a macroscopic object — the target itself provides the “rectification” (reflection).
- Both routes need enormous peak intensity; at PW focus strengths no solid converter survives — plasmas become inevitable (Lecture 4).
1. From “voltage on electrodes” to electromagnetic radiation
Lecture 1 treated RF gaps as places where a voltage is applied. At high frequency this picture is incomplete: any oscillating current on metallic boundaries radiates. A cavity is simultaneously a field shaper and an antenna. As the frequency increases, structures shrink (good) but power leaks into free space (bad) — until the device essentially becomes an optical resonator fed by a laser.
Recap: Widerøe, cyclotron, and why higher frequency helps
Drift-tube length scales as \(d_{\mathrm{drift}} \approx v/(2 f_0) \approx \lambda/2\) for relativistic particles. MHz RF gave metre-scale linacs; GHz microwaves gave centimetre cavities; optical lasers offer micrometre-scale structures.
The cyclotron reuses one RF gap every half turn in a magnetic field (simulator, Exercise 1).
2. Why a plane wave alone cannot accelerate
In a linearly polarized plane wave an electron merely oscillates: \(dv/dt \propto F(t) \propto \sin(\omega t)\), so \(\langle \mathbf{v}\cdot\mathbf{F}\rangle = 0\) over one cycle. This is the essence of the Lawson–Woodward theorem: a free electron in free space cannot extract net energy from a plane EM wave. It is the same transit-time argument as an oversized capacitor gap in Lecture 1.
3. Converter structures: from drift tubes to gratings
Net acceleration requires rectification or phase matching: the particle must see a predominantly accelerating field. Historical solutions include Widerøe drift tubes (shielding during the wrong RF phase) and the Alvarez drift-tube linac (1946), where all tubes sit inside one resonant cavity that enforces the correct TM mode at much higher gradient. Modern microwave linacs use travelling-wave structures; THz and optical versions push the same idea to shorter wavelengths.
For a laser, the analogue is a dielectric grating on a glass substrate: a plane wave enters from below in glass (\(\lambda_{\mathrm{glass}}=\lambda_0/n\), slower phase fronts). In the air grooves the vacuum wavelength \(\lambda_0\) resumes, so groove fronts outrun those still in glass ridges — the origin of the lateral phase slip. Above the grating structure \(\lambda=\lambda_0\) everywhere; only the phase is shifted horizontally. The demo plots \(E_x(x,y,t)\) as a semi-transparent colormap: green where \(E_x>0\), red where \(E_x<0\) (intensity \(\propto |E_x|\)). The electron (white dot) is injected so that it rides the accelerating half-cycle (\(E_x<0\) for \(q=-e\)).
Gap spacing vs Widerøe — why periods lengthen with \(\beta\)
In a Widerøe linac the drift tubes get longer as \(\beta\) increases: with fixed RF frequency \(f\), the particle must spend about half a period in each drift region, so \(L_n \approx \tfrac{1}{2}\beta_n c/f\) grows with \(\beta_n\).
On an optical grating there is no field-free drift like in a Widerøe tube: the electron rides the near-field pattern above the surface in every segment and can be accelerated wherever \(E_x\) points the right way. Air grooves and glass ridges are both part of the active structure — not “gap vs. shielded tube”, but two materials whose optical path (and hence the local phase of \(E_x\)) differs. The ridge retards the wave relative to the groove (\(n>1\)), producing the lateral phase slip that the simulator plots.
One grating period \(\Lambda_n = L_{g,n}+L_{r,n}\) must still place the particle in the accelerating half-cycle when it enters the next period. With fixed laser wavelength \(\lambda\) (frequency \(f=c/\lambda\)), a faster particle covers more distance in one RF cycle, so — as in Widerøe — the period must lengthen with \(\beta\): roughly \(\Lambda_n \sim \tfrac{1}{2}\beta_n\lambda\). If you raise \(v_0/c\) in the demo without retuning \(\Lambda_n\), the electron slips out of phase; lengthening \(L_{g,n}+L_{r,n}\) restores synchronism.
Step 1 — groove height. On the electron track the optical path jumps between groove and ridge by \(\Delta\varphi = k\,h_g\,(n-1)\) with \(k=2\pi/\lambda\). For strong on-axis contrast one first chooses \(h_g\) — e.g. \(\Delta\varphi=\pi\) gives \(h_g = \lambda/\bigl(2(n-1)\bigr)\) (for \(\lambda=0.8\,\mu\)m, \(n=1.5\): \(h_g=0.8\,\mu\)m). The field-coupling factor is \(\eta = |\sin(\Delta\varphi/2)|\).
Step 2 — lengthening periods. With \(h_g\) fixed, choose \(\Lambda_n = L_{g,n}+L_{r,n}\) to grow with \(\beta_n\) (same spirit as Widerøe) and split between groove and ridge. The simulator ships with a equal-period preset (\(h_g=0.8\,\mu\)m, all \(L_{g,n}=L_{r,n}=0.17\,\mu\)m) for \(v_0/c=0.44\), \(E_0=3\) GV/m — increase those lengths together when you raise \(v_0/c\).
Other rectification schemes (reference)
- Disk-loaded waveguides and SW linacs (GHz)
- THz structures (DESY and others)
- Dielectric laser accelerators (DLA; Hommelhoff, SLAC)
4. Light Sail I — the rigid mirror (ideal picture)
Everything in §§1–3 belongs to the conventional accelerator philosophy: a powerful external source creates an oscillating field, and a comparatively small number of charged particles — often essentially one beam — is placed in that field. The converter structure (drift tube, cavity, grating) is engineered separately from the particle; it rectifies the wave so that a single electron or ion can gain energy.
The light sail turns this picture around. The laser still supplies the field, but what is accelerated is not “a few” particles riding an external structure — it is a macroscopic object (a thin foil containing \(\sim 10^{11}\) ions) pushed as a whole. The target is simultaneously the accelerated body and the converter: reflection at its surface rectifies photon momentum into a steady mechanical push. No engineered gap, no half-cycle synchronism — the sail carries its own “rectifier” with it.
This inversion of roles was anticipated remarkably early. Vladimir I. Veksler — who independently discovered phase stability and is co-credited with inventing the synchrotron (1944–45) — later proposed coherent acceleration (mid-1950s): a compact cluster of charges scatters an electromagnetic wave, and the radiation pressure on the cluster accelerates the entire bunch collectively. The scattered force scales as \(N^2\) rather than \(N\) for \(N\) coherently moving electrons — the accelerated object helps generate the very field that pushes it. Modern laser-driven radiation-pressure acceleration and the light sail are direct descendants of that idea. Another simple illustration of the same spirit — Coulomb explosion of a cluster after its electrons are removed — is developed in Exercise 2, Part D.
In the ideal picture below we deliberately ignore ionization, transmission, absorption, and heating. The target is a perfect mirror of mass \(M\): every reflected photon pushes the entire rigid object forward. Ions are assumed bound to the electrons and carried along. This thought experiment is the simplest answer to “how could a laser pulse accelerate matter?” — and the starting point for radiation-pressure acceleration (RPA) later in the course.
The two philosophies contrast sharply:
- Phase-matched converter (Widerøe, grating, …): external structure rectifies the wave; a few particles extract net work from an oscillating field.
- Light sail / coherent acceleration (Veksler, …): the accelerated object itself reflects (rectifies) photon momentum; a macroscopic body is pushed as one unit.
5. Photon momentum and radiation pressure
Each photon of energy \(E\) carries momentum \(p = E/c = h/\lambda\). Upon reflection the momentum transfer to the mirror is \(\Delta p = 2h/\lambda\) (reversal of the normal component). For a continuous beam of power \(P\) with reflectivity \(R\), the force is
\[ F = \frac{2 R P}{c}. \]
This is the same result one obtains from the Maxwell stress tensor for a perfectly reflecting surface. The push acts for the entire pulse duration — unlike a grating gap, there is no need to arrive at a particular RF phase.
6. Equation of motion and a worked example
For a pulse with constant power \(P\) during the interaction, \(\mathrm{d}p/\mathrm{d}t = 2RP/c\). If the sail is non-relativistic and the pulse energy is \(E_L = \int P\,\mathrm{d}t\), a simple estimate gives
\[ \beta = \frac{v}{c} \approx \frac{2 E_L}{M c^2}. \]
- Pulse energy \(E_L = 30\,\mathrm{J}\) (order of magnitude for a compressed ATLAS shot)
- Water-density foil \(\rho\approx 10^3\,\mathrm{kg\,m^{-3}}\), illuminated spot of diameter \(5\,\mu\mathrm{m}\) (\(A=\pi(2.5\,\mu\mathrm{m})^2\approx 2\times 10^{-11}\,\mathrm{m^2}\))
- Choose thickness so that \(\beta=0.1\): from \(\displaystyle\beta\approx\frac{2E_L}{M c^2}\) one needs \(M c^2=2E_L/\beta=600\,\mathrm{J}\), hence \(M\approx 6.7\times 10^{-15}\,\mathrm{kg}\)
- With \(M=\rho A d\) that sets \(\displaystyle d=\frac{M}{\rho A}\approx 0.3\,\mu\mathrm{m}\)
The scaling shows why thin, light targets are attractive: halving the mass doubles the velocity. At the same time, the sail must be thick enough to stay reflective — a tension developed fully in Lecture 9 (Light Sail II on plasma).
7. Demos and history
Light pressure is not an exotic laser effect. James Clerk Maxwell predicted it from electromagnetic theory in his Treatise on Electricity and Magnetism (1873); Ludwig Boltzmann and Adolfo Bartoli arrived at the same conclusion independently from thermodynamic arguments (1876). The first experimental demonstration was by Pyotr N. Lebedev (1900, published 1901) using a torsion balance with metal vanes in a high vacuum — a genuine radiometer experiment, not the Crookes “light mill” sold as a curiosity, whose spin is dominated by residual-gas heating rather than photon momentum. See Lebedev, Untersuchungen über die Druckkräfte des Lichtes, Ann. Phys. 311, 433 (1901) (setup conceptually close to a torsion radiometer; the classroom light mill is the misleading cousin of that apparatus).
The electromagnetic analogue of a charged capacitor is still useful pedagogy: two plates carrying opposite charge attract each other — macroscopic momentum exchange in the static limit.
- Maxwell (1873) / Bartoli (1876): theoretical prediction of radiation pressure from electromagnetism and thermodynamics.
- Lebedev (1901): first quantitative measurement of radiation pressure on solids, confirming Maxwell–Bartoli (Ann. Phys. 311, 433, 1901; announced Paris 1900).
- Nichols & Hull (1901): independent confirmation with a similar torsion-balance radiometer (Dartmouth).
- Cook, Flowers & Arnold (1962): first measurement of laser output by light pressure (Proc. IRE 50, 1693).
- Marx (1966): theoretical proposal of an interstellar vehicle propelled by a terrestrial laser beam — relativistic light-sail equations (Nature 211, 22).
- Breakthrough Starshot / StarChip (2016+): gram-scale sail pushed by a ground-based laser array — the same \(F=2P/c\) physics at astronomical scale.
Capacitor analogy (qualitative)
In Lecture 1 we accelerated with a voltage between plates. Here the “voltage” is supplied by the laser field and the recoil is momentum conservation rather than energy gain of a single charge in a gap. Both pictures illustrate that electromagnetic fields carry momentum as well as energy.
8. Peak intensity — and why no material survives
Shorter wavelength alone is not enough — we also need enormous peak power (not average power: CPA delivers Joules in tens of femtoseconds). Modern systems reach petawatt levels (e.g. ATLAS 3000 at CALA, LMU Munich), far beyond pulsed microwave sources (~100 GW). For oscillatory motion in a field, \(E_{\mathrm{kin}} \propto I_0 \lambda^2 \propto P\): peak power sets the energy scale extractable from the wave — whether by phase matching or by radiation pressure.
Focussing that power to a few micrometres gives intensities of order \(10^{21}\,\mathrm{W\,cm^{-2}}\). The corresponding field amplitudes exceed atomic binding fields (\(\sim 10\,\mathrm{GV\,m^{-1}}\)): any solid target is field-ionized and turned into a plasma before the main pulse peak arrives. So the grating of §3 and the rigid mirror of §§4–7 are both thought experiments on solids — experimentally we always deal with plasmas. That is the bridge to Lecture 4 (ionization and why the rigid mirror fails).
Derivation sketch: \(E_k \propto I_0\lambda^2\)
From \(m\,\mathrm{d}v/\mathrm{d}t = q E_0\cos\omega_0 t\) one finds \(E_k = q^2 E_0^2/(2m\omega_0^2)\) and, using \(I_0 = \tfrac{1}{2}\varepsilon_0 c E_0^2\), the proportionality \(E_k \propto I_0 \lambda^2\).
9. What is missing?
The light sail of §§4–7 and the grating of §3 are both idealised solids. PW-class lasers field-ionize any real foil long before the pulse peak: free electrons appear, the surface ceases to be a perfect mirror, and energy is absorbed rather than reflected. Lecture 4 asks where the free carriers come from and what replaces the rigid mirror — the path to laser–plasma acceleration.
Work through Exercise 2 (peak intensity and material limits) before the tutorial.