Laser-Ion Acceleration

1. One equation for every ideal lens

For a particle moving mainly along \(z\), the paraxial transverse equation is \(pv\,x''=F_x\). If \(F_x=-\kappa x\), then \[ \boxed{x''+Kx=0},\qquad K=\frac{\kappa}{pv},\qquad k=\sqrt{K}. \] A constant-\(K\) element of length \(L\) has transfer matrix \[ M=\begin{pmatrix}\cos kL & k^{-1}\sin kL\\ -k\sin kL & \cos kL\end{pmatrix}, \qquad \mu=kL. \]

Magnetic rigidity \(B\rho=p/q\) sets the energy scale. For \(\sim 15\,\mathrm{MeV}\) protons, \(B\rho\simeq 0.56\,\mathrm{T\,m}\).

2. Watch the trajectories

Switch between quadrupole (F solid / D dashed), solenoid (round + rotation), and active plasma lens (round, no DOFO). Same \(E\), aperture, and length — different \(K\).

(Open full page. Defaults use a pedagogical field strength; retune \(B\), \(a\), \(L\) in Advanced.

3. Quadrupole — strong, but one plane at a time

Near the axis, \(B_y=Gx\), \(B_x=Gy\) with \(G=B_{\mathrm{edge}}/a\). Then in the focusing plane \[ K_q=\frac{B_{\mathrm{edge}}}{a\,B\rho}, \] while the other plane has \(-K_q\) (hyperbolic solutions). Maxwell forbids a static magnetic lens that focuses both planes at once with a pure quadrupole field.

Hence the doublet \(F\!-\!D\) (and \(D\!-\!F\) in the other plane) — see the Lecture 13 doublet simulator. For large-divergence laser ions the intermediate envelope growth is the DOFO penalty: both magnets must clear the enlarged beam.

4. Solenoid — why \(B_z\) focuses at all

A particle exactly parallel to a uniform \(B_z\) feels no Lorentz force. Focusing comes from the fringe fields:

1. Entrance fringe: radial \(B_r\) imparts azimuthal momentum.
2. Inside: \(v_\varphi\times B_z\) provides the restoring force (Larmor rotation).
3. Exit fringe: removes (most of) the azimuthal motion; a net focusing kick remains.

In the co-rotating Larmor frame one recovers again \(x_L''+K_s x_L=0\) with \[ \boxed{K_s=\Bigl(\frac{B_s}{2B\rho}\Bigr)^2},\qquad \frac1{f_s}=\frac{B_s^2 L}{4(B\rho)^2},\qquad L_{\pi/2,s}=\frac{\pi B\rho}{B_s}. \] Both planes focus equally; the transverse phase-space image also rotates (visible in the end-view panel).

5. Active plasma lens (and lithium lens)

A longitudinal current \(I\) with uniform \(J_z\) in a cylinder of radius \(a\) produces \[ B_\varphi(r)=\frac{\mu_0 I}{2\pi a^2}\,r = g r,\qquad B_{\mathrm{edge}}=\frac{\mu_0 I}{2\pi a}. \] The Lorentz force is radial and linear: \[ \boxed{K_{\mathrm{APL}}=\frac{B_{\mathrm{edge}}}{a\,B\rho}=K_q}. \] Same local \(K\) as a quadrupole focusing plane — but simultaneously in every azimuth. An ideal lithium lens shares this field class; differences are engineering (conductor vs discharge).

Ampère’s law forbids “freezing” \(B_\varphi\propto r\) into a current-free permanent-magnet bore: \((\nabla\times\mathbf{B})_z=2g\neq 0\) requires axial current in the aperture. That is what makes APL/Li lenses special — and technically hard (homogeneous \(J_z\), timing, heat).

6. Head-to-head: APL vs solenoid

Equal \(K\) implies \[ \boxed{B_{\mathrm{APL}}=\frac{B_s^2 a}{4\,B\rho}}. \] Example: \(B_s=8\,\mathrm{T}\), \(a=4\,\mathrm{cm}\), \(B\rho=0.56\,\mathrm{T\,m}\) \(\Rightarrow B_{\mathrm{APL}}\simeq 1.1\,\mathrm{T}\). Equal numerical field \(B\) instead gives \(K_{\mathrm{APL}}/K_s=4B\rho/(aB)\sim\mathcal{O}(10)\) for these numbers — optics only, not a claim that 8 T azimuthal edge field is easy.

7. What “acceptance” means

• Geometric: \(\theta_{\mathrm{geom}}\lesssim a/d\) for a source at distance \(d\).
• Thin-lens edge kick: \(a/f\) (or \(D/f\) with \(D=2a\)).
• Thick quarter-wave collection from a point source at the entrance: \(\theta_{\max}\simeq a\sqrt{K}\) (\(\sqrt{a B_{\mathrm{edge}}/B\rho}\) for APL; \(a B_s/(2B\rho)\) for solenoid).

For laser ions, \(d\) is often the scarce resource: every millimetre before the first optic burns geometric acceptance. Hence interest in source-integrated focusing.

8. Compact comparison

9. Outlook

Same harmonic backbone also covers Gabor lenses (\(E_r\propto -r\)), passive/self-generated plasma focusing, helical/travelling-field structures, and source-integrated schemes. For the core course it is enough to remember: technology changes \(K\); laser ions force you to ask which acceptance and which energy slice you actually need.