Beam optics

The transverse transport of a charged-particle beam relative to a reference trajectory. Linear focusing gives betatron oscillations, described by Hill's equation and Courant-Snyder parameters; the beam envelope depends on its geometric emittance.

A linear paraxial map between transverse position and trajectory angle at two beamline locations. A drift of length $d$ has $M=\left(\begin{array}{cc}1& d\\ 0& 1\end{array}\right)$; a focusing thin quadrupole has $M=\left(\begin{array}{cc}1& 0\\ -1/f& 1\end{array}\right)$. Lossless uncoupled maps have determinant one. Repeated-cell bounded oscillatory motion requires $|\mathrm{tr}M|<2$; at equality a Jordan block can still cause secular growth. This beam-optical map is distinct from a combinatorial transfer matrix.

For an initially parallel bundle passing through opposite equal thin-quadrupole strengths separated by $L$, the two downstream focal distances differ as shown. Approximating the doublet by a common effective focal length $f^2/L$ discards that difference. A real common final focus requires additional incoming optical information or adjusted strengths.

A magnetic chicane's momentum-dependent path length does not specify a compression factor from initial length and marginal energy spread alone. A suitable longitudinal position-energy correlation is required. With zero correlation the variance above is minimized at $R_{56}=0$, so such a chicane cannot shorten that ideal uncorrelated bunch. An imposed chirp and its uncorrelated residual determine the achievable minimum.

A lattice cell with two bending magnets whose dispersion and dispersion slope vanish at its boundaries. In an ideal identical-uniform-bend model with horizontal damping partition one, its minimum geometric emittance is $C_q{\gamma }^2{\theta }^3/(4\sqrt{15})$, larger than the unconstrained theoretical minimum because of the achromatic boundary conditions.

The derivative of transverse betatron tune $Q$ with respect to relative momentum error $\delta$. For a thin-lens FODO cell with fixed quadrupole fields, natural cell chromaticity is $-\tan (\mu /2)/\pi$ on the stable phase-advance branch. Zero corrected chromaticity requires additional momentum-dependent focusing.

A beam intensity or photon flux per transverse phase-space volume, with any spectral bandwidth convention stated. In the far-from-diffraction-limited case with fixed flux and negligible dispersion contributions, brightness scales inversely with ${\epsilon }_x{\epsilon }_y$.

Photon source widths combine electron and diffraction widths in quadrature. At fixed optics, reducing electron emittance eventually has little effect when both radiation size and divergence dominate. This criterion differs from comparing emittances alone if the optical beta function is not matched to the diffraction ellipse.

A repeating focusing-drift-defocusing-drift lattice cell, with opposite quadrupole focusing in the two transverse planes. Stability and phase advance depend on focal strengths and separations.

The positive function $\beta (s)$ and the displayed associated parameters describe a stable linear beam ellipse. A trajectory with invariant $ϵ$ obeys $ϵ={\gamma }_T{z}^2+2{\alpha }_{T}z{z}^{\prime }+\beta z^{\prime 2}$ and has phase advance ${\varphi }^{\prime }=1/\beta$. Writing $\omega =\sqrt{\beta }$ gives ${\omega }^{\prime \prime }+k(s)\omega =1/{\omega }^3$; the coefficient of $k$ is $\omega$, not one.

The area occupied by a beam in transverse position-momentum or position-angle phase space, with a stated area or rms convention. Geometric emittance uses trajectory angle, while normalized emittance removes the ideal kinematic damping due to acceleration.

The phase-space area measure in the vertical transverse plane. If coupling is described by $k={\epsilon }_y/{\epsilon }_x$, changing the coupling and changing the natural emittance both alter its equilibrium value.

The phase-space area measure of the horizontal transverse beam distribution. With negligible dispersion and a specified optical function, it determines beam size and angular spread.

The equilibrium horizontal beam emittance set by synchrotron-radiation excitation and damping in an uncoupled storage ring. The radiation-integral relation is ${\epsilon }_{x0}=C_q{\gamma }^2{I}_5/(J_x{I}_2)$; similar cell optics give a cubic dependence on bending angle.

The geometric emittance multiplied by the relativistic velocity and Lorentz factor. For paraxial acceleration without dilution, this is proportional to transverse canonical phase-space area and remains constant. Thus geometric emittance decreases approximately as inverse beam momentum.

For a centered beam, the displayed rms area in $(z,z^{\prime })$ coordinates. A matched beam has $⟨z^2⟩=ϵ\beta$, where $\beta$ is one of the Courant-Snyder parameters. Under ideal acceleration, adiabatic damping of geometric emittance decreases this area even when the canonical phase-space area is preserved.

For slowly accelerated paraxial motion, $z^{\prime }\simeq p_z/p_s$ decreases as longitudinal momentum grows while transverse canonical phase-space area is conserved. The normalized emittance stays constant in the ideal limit, so the geometric emittance scales inversely with longitudinal momentum. This is kinematic damping rather than dissipation of canonical area.

Small transverse oscillation about an accelerator's reference orbit. In a linear periodic lattice the displacement obeys Hill's equation $z^{\prime \prime }+k(s)z=0$.

For rms geometric beam emittance $ϵ$, the transverse beam variance is ${\sigma }^2(s)=ϵ\beta (s)$. At a beam waist, the angular variance is $ϵ/\beta$. Reducing both transverse amplitude functions by a factor $a$ at fixed emittances increases the ideal head-on collider luminosity by $a$; practical finite-bunch-length and crossing-angle effects can limit this gain.

Alternating focusing and defocusing quadrupole magnets, separated by drifts, can give stable net focusing in both transverse planes. Stability depends on the full transfer matrix of the lattice cell, not on the sign of one magnet alone.

A magnet whose transverse field varies linearly with displacement. Its Lorentz force focuses a charged-particle beam in one transverse plane and defocuses it in the other. Oppositely oriented elements provide alternating-gradient focusing.

At fixed gradient and magnet length, the signed thin-lens focal length is proportional to particle momentum. For total energy error $\delta =(E-E_0)/E_0$, differentiating $p^2{c}^2=E^2-m^2{c}^4$ gives $d\ln p/d\ln E=1/{\beta }^2$. The ultrarelativistic result is $f=f_0(1+\delta )$ to first order. If energy means kinetic energy instead, the coefficient is $(K+m{c}^2)/(K+2m{c}^2)$.

An element of the strong focusing system near a collider interaction point. It helps form a small transverse beam waist there, matched to the surrounding periodic lattice. Small collision-point beta functions generally imply large envelopes and strong chromatic effects in nearby magnets.