Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2016 / S19

Let $x_i=L/f_i$, with positive $f$ focusing in the first transverse plane. The doublet matrix is $M=Q_{2}D(L)Q_1$, with $Q_i=\left(\begin{array}{cc}1& 0\\ -1/f_i& 1\end{array}\right)$. Its lower-left entry is $C_x=-(x_1+x_2-x_1{x}_2)/L$. In the other plane both quadrupole signs reverse, giving $C_y=-(-x_1-x_2-x_1{x}_2)/L$. Thus positive effective focusing power in both planes requiresFor a parallel incident ray bundle, actual downstream focal distances are ${\ell }_x=L(1-x_1)/(x_1+x_2-x_1{x}_2)$ and ${\ell }_y=L(1+x_1)/(-x_1-x_2-x_1{x}_2)$. Requiring both foci downstream adds $|x_1|<1$. Positive effective power alone need not imply a real image after the final element. The original schematic below distinguishes these conditions.

The beam rigidity is $B\rho \simeq p/(0.2998\mathrm{GeV}/(c\mathrm{T}\mathrm{m}))\simeq 1.67\times 10^5$ T m. Equal opposing strengths have $f=B\rho /(G{L}_q)$, and weak separated-doublet net power is $1/F\simeq L/f^2$. Taking $F\sim L^{\ast }=50$ m gives the design familyThere is not a unique pair $(L_q,L)$ without a spacing/layout choice. A compact estimate $L\sim L_q$ gives about 18 m for each, showing immediately that neither is extremely small compared with 50 m and the advertised asymptotic approximation is only rough. For example $L=30$ m gives $L_q\sim 14$ m in the same estimate.

An equal-strength doublet also has exact parallel-input focal distances $f^2/L-f$ and $f^2/L+f$: it is astigmatic, so a single exact common 50 m focus cannot be imposed without specifying incoming optical functions or allowing unequal strengths. The family above is a scale estimate of net focusing, not a complete collider final-focus design.

A uniform nonrelativistic electron column has radial electric field $E_r=-n_{e}er/(2{ϵ}_0)$ inside its radius. A relativistic proton therefore has focusing strength $K=e^2{n}_e/(2{ϵ}_{0}pv)$ per unit length. Electron-current magnetic effects are small under the stated nonrelativistic assumption. To change the nominal doublet strength $1/L^{\ast }$ by $10^{-4}$ requiresOne illustrative choice is a one-metre lens with $n_e\simeq 1.1\times 10^{16}$ m${}^{-3}$; other density-length combinations work. The electron beam must cover the proton beam and have a sufficiently uniform profile. Its placement changes the detailed transfer-map correction; the estimate treats the required additional integrated focusing at the entrance and does not claim a uniquely specified electron-lens engineering design.

At zero dispersion and $\alpha =0$, $\sigma =\sqrt{ϵ\beta }$ and ${\sigma }^{\prime }=\sqrt{ϵ/\beta }$. The results areThe repeated $D_x,D_y$ entries in the source are interpreted as intended vanishing dispersion and its slope. If dispersion slopes were not zero, energy spread would contribute to divergence and further data would be needed.

State a radiation-width convention. For rms Gaussian-equivalent single-electron undulator radiation take ${\sigma }_r=\sqrt{\lambda L}/(4\pi )$ and ${\sigma }_r^{\prime }=\sqrt{\lambda /L}$, so the diffraction emittance is $\lambda /(4\pi )$. These give 11.3 micrometres and 70.7 microradians. Convolve independent electron and radiation widths in quadrature. The total photon source is thenSome approximate undulator width conventions differ by order-one factors; these must be kept consistent in the following brightness comparison.

At fixed flux and bandwidth, brightness is inversely proportional to ${\Sigma }_x{\Sigma }_x^{\prime }{\Sigma }_y{\Sigma }_y^{\prime }$. Lowering both geometric emittances by ten makes the electron rms widths and divergences smaller by $\sqrt{10}$. Add the unchanged radiation widths in quadrature again and divide the old phase-space product by the new one. The resulting brightness factor is approximately $\boxed{3.87}$, an increase of about $287\%$, rather than the factor 100 that would follow if diffraction were negligible in both planes. The exact value under the adopted width convention is given by the accompanying check and plot.

Emittance reduction has little effect when radiation dominates every source width and divergence. At fixed $\beta =L=2$ m, the strongest condition is $\lambda L/(16{\pi }^2)\gg {ϵ}_x\beta$, or $\boxed{\lambda \gg 4.7\times 10^{-7}\mathrm{m}}$. The divergence condition only requires $\lambda \gg {ϵ}_{x}L/\beta =3$ nm. There is no sharp “negligible” cutoff without an allowed percentage. If the optics were reoptimized to match diffraction instead, the weaker comparison $\lambda /(4\pi )\gg {ϵ}_x$ would set the relevant scale; it is not the same fixed-beta problem.

For the same lattice radiation-integral ratio, natural geometric emittance scales as ${\gamma }^2$. A factor-ten decrease therefore requires $\boxed{E^{\prime }=3/\sqrt{10}\simeq 0.949\mathrm{GeV}}$. Keeping coupling ratio fixed reduces the vertical emittance by the same factor, within that ideal scaling model.

Lower energy changes the source spectrum as well as emittance. At fixed bending geometry, bending-radiation critical energy scales as ${\gamma }^3$, so it falls by $10^{3/2}$; for fixed undulator period and deflection parameter the fundamental wavelength grows as ${\gamma }^{-2}$, by ten. A smaller electron emittance therefore does not automatically improve brightness at the original 10 nm wavelength: the device may no longer radiate there on its fundamental, flux and harmonic operation change, and diffraction remains. Lattice redesign can retain beam energy and target spectrum while reducing emittance, which is why energy reduction alone is not a generally satisfactory substitute.

TODO — the source does not supply the longitudinal position-energy correlation, imposed chirp, or chicane dispersion/geometry needed for a unique compressed bunch length. A four-bend drawing and an uncorrelated energy spread are insufficient. In linear optics,If the given spread is uncorrelated, the minimum is the unchanged 0.2 m, not a compressed submillimetre bunch. If a chirp is supplied, optimal $R_{56}$ depends on it and on the uncorrelated residual spread.

A sometimes intended phase-space lower-bound estimate conserves longitudinal emittance and permits final relative energy spread of order unity, giving ${\sigma }_{zf}\gtrsim {\sigma }_z{\sigma }_{\delta }\sim 20\mu$m. This is an additional extreme energy-spread allowance, not a consequence of an unspecified magnetic chicane. If one illustratively matches a plasma scale $k_p{\sigma }_{zf}\sim 1$, then $n_e\sim {ϵ}_0{m}_e{c}^2/(e^2{\sigma }_{zf}^2)\simeq 7\times 10^{22}$ m${}^{-3}$. Matching to a whole wavelength instead changes this by $4{\pi }^2$. The parametric density relation is valid; the unique requested numerical compression is not established by the source.

The required energy increment is 9.9 TeV. If the usable longitudinal gradient is $\eta E_0$, with $E_0=m_{e}c{\omega }_p/e$, the channel length isUnder the illustrative preceding ${\sigma }_{zf}=20\mu$m and $k_p{\sigma }_{zf}=1$ choices, $E_0\simeq 25.6$ GV m${}^{-1}$ and $L_p\simeq 390/\eta$ m. TODO — a unique numerical length inherits the missing compression/density specification and additionally requires a usable wake amplitude/loading factor $\eta$ or driver information. A driver energy per proton alone does not fix its bunch population or accelerating gradient.

Propagation through an already formed cold plasma requires ${\omega }_L>{\omega }_p$, hence $\boxed{{\lambda }_L<{\lambda }_p=2\pi c/{\omega }_p}$. For the illustrative density above, that is a cutoff near 126 micrometres. It is a propagation ceiling, not a sufficient condition for producing ionization.

TODO — the unique cutoff inherits the unspecified density, and a literal maximum wavelength for gas ionization also needs gas species, laser intensity and the ionization mechanism. A single-photon threshold would impose $hc/\lambda \ge I_{\mathrm{ion}}$; strong-field or multiphoton ionization is a different criterion. None of those data is supplied. The density-dependent propagation condition is the part determined symbolically.

For a 1 micrometre laser, $n_c\simeq 1.1\times 10^{27}$ m${}^{-3}$ and a cold underdense wave has ${\gamma }_{\varphi }^2\simeq n_c/n_e$. At peak wave scale $E_0$, optimal passage through an accelerating half-cycle gives ideal stage gain $\Delta \mathcal{E}\sim 4(n_c/n_e)m_e{c}^2$; crest-to-zero injection gives half that. With the illustrative $n_e\simeq 7\times 10^{22}$ m${}^{-3}$, the optimal ideal gain is about 32 GeV and a 9.9 TeV increment needs order 300 stages, or order 600 with the crest convention.

TODO — no unique stage count follows without the missing density and laser/wake amplitude assumptions. Depletion, diffraction, guiding, beam loading, injection, energy spread, driver evolution and interstage losses can all increase or otherwise alter the requirement. The parametric count is $N\gtrsim \Delta {\mathcal{E}}_{\mathrm{total}}/[4\eta (n_c/n_e)m_e{c}^2]$, with the phase and field conventions explicitly retained.