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 / 2

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.