13.12 CBSCAT—Compton back-scattering of a counter-propagating laser off the beam.

Compton back-scattering of a counter-propagating laser off the beam.
Parallel capable? : yes
GPU capable? : no
Back-tracking capable? : no
Spin-tracking capable? : no

Compton back-scattering of a counter-propagating laser off the beam.

This element models Compton back-scattering of a counter-propagating (or crossing-angle) laser pulse off the electron beam, following and extending Pan et al. [2]. It is a zero-length, per-particle stochastic element: on each pass every particle has a probability of undergoing a single Compton scatter, and if it does its transverse angles and fractional momentum deviation are changed according to Eq. (4) of that reference. The element may optionally record the emitted (back-scattered) photons to an SDDS file for γ-source spectrum studies.

Recoil model. Let γ and β be the Lorentz factor and velocity of the individual electron (computed from its own momentum P = (1 + δ)P₀, not the reference value, so a beam with energy spread scatters with the correct per-electron kinematics), E_e = γm_ec² its energy, E_h = hc∕λ the lab-frame laser-photon energy (set by LASER_WAVELENGTH), and θ₀ the collision angle between the electron and laser directions (COLLISION_ANGLE; θ₀ = π is head-on). Define the rest-frame incident photon energy

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For a scatter with photon polar angle θ (measured from the electron direction) and azimuth ϕ, the coordinate changes applied are

The Eq. (4) mapping above is a linearized lab-frame recoil: the transverse terms are the paraxial projection x′ = p_x∕p_z, and the energy change is evaluated using the incident rest-frame photon energy E′. It is therefore exact only in the Thomson limit ϵ ≡ E′∕(m_ec²) → 0; for ϵ not small it overestimates the electron energy loss because it omits the rest-frame Compton down-shift. Setting EXACT_RECOIL= 1 replaces this mapping with an exact, per-particle treatment: the electron and incident photon four-momenta are boosted into the electron rest frame (boost along the electron’s own direction), the photon is scattered there with exact Compton energy–momentum conservation,

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and the outgoing electron and photon four-momenta (formed by conservation) are boosted back to the lab, where the electron momentum is mapped to (x′,y′,δ) without approximation. This matches the analytic Compton edge 4γ²E_h∕(1 + 2ϵ) and should be used in the hard inverse-Compton / γ-ray regime (ϵ ≳ 0.1); at small ϵ it agrees with the default to a few percent. The overlap integral and the sampling of θ are identical in both modes. The default (EXACT_RECOIL= 0) retains the Eq. (4) mapping for backward compatibility.

Cross section. The scattered-photon polar angle is sampled from the differential cross section selected by CROSS_SECTION:

• "thomson": dσ∕dΩ ∝ (1 + cos²θ), with total cross section σ_T = r_e² [49].
• "klein-nishina" (default): the full Klein–Nishina expression [50] with ϵ = E′∕(m_ec²), and the corresponding Klein–Nishina total cross section.

In the keV rest-frame-energy regime typical of optical-laser Compton sources the two choices agree to order ϵ; both are provided for generality.

Scatter probability (luminosity overlap). The mean number of scatters for a given particle on a given pass is computed as a luminosity-overlap integral of that particle’s path through the Gaussian laser pulse,

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integrated over the electron beamline coordinate z, where n_ph is the laser photon density, (1 -β cosθ₀) is the relative-velocity (flux) factor — exact for any angle — and the 1∕β converts the time integral to a path-length integral. The photon number is N_ph =PULSE_ENERGY∕E_h unless overridden by N_PHOTONS.

The overlap is evaluated in the laser propagation frame rather than the beamline frame, so the geometry is correct for any collision angle, including a 90^∘ side collision. The propagation direction and two transverse laser axes are built from COLLISION_ANGLE (θ₀, the angle between +ẑ and the laser propagation; θ₀ = π is head-on) and COLLISION_AZIMUTH (ϕ₀, measured from + toward +ŷ, so ϕ₀ = 0 is a crossing in the horizontal x–z plane and ϕ₀ = π∕2 a vertical crossing):

• Hourglass: the laser rms spot size grows away from the waist along the propagation direction as σ(s) = σ₀, where s is the distance from the waist measured along , with Rayleigh range z_R = 4πσ₀²∕λ computed separately for SIGMA_X and SIGMA_Y from LASER_WAVELENGTH. The waist location is set by FOCUS_POSITION (S₀, along ẑ) together with the transverse offsets DX, DY.
• Crossing geometry: the electron’s separation from the moving pulse center is projected onto (,₁,₂), so both the reduced interaction length of an oblique collision and the exact flux factor (1 - β cosθ₀)∕β are captured.

Each particle’s true (velocity-corrected) arrival time, referenced to the bunch centroid and offset by LASER_DELAY (a time in seconds), sets its overlap, so head/tail and off-axis particles sample different, hourglass-broadened overlaps. Because the exponent is quadratic in z, the quadrature is centered on the overlap peak and spans ±5 effective sigma about it, evaluated over N_STEPS points (the full hourglass-broadened density is used at each point). A single scatter per pass is applied (valid for μ « 1); if μ > 1 it is clamped to 1 and a warning is issued. The overall rate can be scaled with FACTOR.

Photon output. If PHOTON_OUTPUT_FILE is given, the emitted back-scattered photons are written to an SDDS file with columns Ep (photon energy, eV), x, xp, y, and yp. Photon output is available in the serial version of elegant only.

The output filename may be an incomplete filename. This means it may contain one instance of the string format specification “%s” and one occurence of an integer format specification (e.g., “%ld”). elegant will replace the former with the rootname (see run_setup) and the latter with the element’s occurrence number. This allows labeling the files from different interaction points with sequential indices.

Notes. The element applies at most one Compton event per particle per pass, consistent with Eq. (4). For multi-pass (storage-ring) use, STARTONPASS and ENDONPASS gate the passes on which scattering occurs, and PASS_INTERVAL further restricts scattering to every PASS_INTERVAL-th pass counting from STARTONPASS (the default of 1 acts on every pass in the window).