Appendix F — OPALX Design Note: Laser–Electron ICS and gamma-gamma Collision Physics

F.3.1 Manual-visible Anchor Points

The current manual states that:

• TRACK in OPALX accepts BEAM or BEAMS, with BEAMS taking precedence
• the public BEAMBEAM documentation still describes the mirrored-bunch prototype and marks the copy sketch as not yet implemented
• BEAM provides predefined charged species such as ELECTRON and POSITRON, but no predefined PHOTON
• ScatteringPhysics in particle-matter interactions already uses internal substeps when the outer step is too large

F.3.2 Runtime Anchor Points

The natural insertion point for this work is:

• tracker-owned lifecycle and lab-frame control in the tracking layer
• bunch-owned field-mesh and scatter mechanics in the bunch layer
• diagnostics already separated from the core state
• HDF5 writers already available for per-step diagnostics

F.6.1 Suggested Input Syntax

// Electron beams
EL: BEAM, PARTICLE = ELECTRON, PC = P0L, NPART = nL, BFREQ = rf, BCURRENT = iL, SOURCES = srcL;
ER: BEAM, PARTICLE = ELECTRON, PC = P0R, NPART = nR, BFREQ = rf, BCURRENT = iR, SOURCES = srcR;

// Photon beams created by ICS
GL: BEAM, PARTICLE = PHOTON, NPART = 0;
GR: BEAM, PARTICLE = PHOTON, NPART = 0;

// Optional e+/e- output beams from gamma-gamma events
EP: BEAM, PARTICLE = POSITRON, NPART = 0;
EM: BEAM, PARTICLE = ELECTRON, NPART = 0;

// Analytic laser pulses
LASL: LASER, MODEL = GAUSSIAN,
      WAVELENGTH = 1.03e-6,
      PULSEENERGY = 1.0,
      PULSELENGTH = 2.0e-12,
      WAISTX = 5.0e-6, WAISTY = 5.0e-6,
      DIR = {0, 0, -1},
      STOKES = {0, 0, 1},
      PROFILE = GAUSS;

// Discrete physics handlers
ICSL: COLLISIONPHYSICS, TYPE = LASERCOMPTON,
      LASER = LASL,
      PRODUCTBEAM = GL,
      MODEL = KLEINNISHINA,
      POLARIZATION = TRUE,
      PMAX = 0.05,
      MODE = WEIGHTED_SPLIT;

GGLUM: COLLISIONPHYSICS, TYPE = GGLUMINOSITY,
       BEAMS = {GL, GR},
       NBIN = 200,
       WMIN = 1.0e6,
       WMAX = 5.0e11,
       HELICITY = TRUE;

BW: COLLISIONPHYSICS, TYPE = GGBREITWHEELER,
    BEAMS = {GL, GR},
    PRODUCTS = {EM, EP},
    PMAX = 0.02,
    MODE = WEIGHTED_EVENT;

LBL: LASERBEAM, ELEMEDGE = s_conv_l, L = 0.003, PHYSICS = ICSL;
IP1: BEAMBEAM, ELEMEDGE = s_ip, L = 0.002, PROCESSES = (GGLUM, BW);

F.7.1 Photon Species

A photon macro-particle should carry at least:

• position
• direction unit vector
• energy
• weight w
• Stokes vector
• creation time
• parent ID and generation index
• alive flag

Photons drift at c, do not deposit charge into space-charge solvers, and must participate in diagnostics.

F.7.2 Variable Macro-Particle Weights

Weighted reaction modes are essential because the event probabilities per substep are typically small. Every particle species therefore needs a floating-point weight.

F.7.3 Polarization Metadata

The proposed storage is the Stokes vector

\[ \mathbf S = (S_0, S_1, S_2, S_3), \qquad S_0 \ge 0, \qquad S_1^2 + S_2^2 + S_3^2 \le S_0^2. \tag{F.1}\]

For circular-helicity luminosity, S_3 is the key quantity.

F.8.1 Suggested Class Layout

Public objects:

• Laser.h, LaserRep.h, OpalLaser.h
• LaserBeam.h, LaserBeamRep.h, OpalLaserBeam.h
• CollisionPhysics.h, CollisionPhysicsRep.h, OpalCollisionPhysics.h

Runtime / process layer:

• DiscreteCollisionProcess
• LaserComptonProcess
• GammaGammaLuminosityProcess
• GammaGammaBreitWheelerProcess

Diagnostics:

• H5LaserBeamDiagnosticsWriter
• H5GammaGammaDiagnosticsWriter

F.8.2 Process Interface

class DiscreteCollisionProcess {
public:
    virtual double suggestDtMax(const CollisionContext& ctx) const = 0;
    virtual void apply(CollisionContext& ctx, double dt_sub) = 0;
    virtual void writeDiagnostics(const CollisionContext& ctx) = 0;
    virtual ~DiscreteCollisionProcess() = default;
};

F.8.3 Suggested Tracker Call Pattern

advanceOneGlobalStep(dt):
    advanceExternalAndCollectiveFields(dt)

    for each active collision element:
        ctx = buildCollisionContext(...)
        dt_sub = min_over_processes(process.suggestDtMax(ctx))
        n_sub = ceil(dt / dt_sub)
        dt_sub = dt / n_sub

        repeat k = 1 .. n_sub:
            for process in active_processes:
                process.apply(ctx, dt_sub)

        for process in active_processes:
            process.writeDiagnostics(ctx)

F.9.1 Photon Transport

For version 1, photons free-stream between discrete interactions:

\[ \frac{d\mathbf x}{dt} = c\,\hat{\mathbf n}, \qquad \frac{d\hat{\mathbf n}}{dt} = 0, \qquad \frac{dE_\gamma}{dt} = 0. \tag{F.2}\]

F.9.2 Analytic Laser Model

The note proposes a Gaussian pulse model with local power density

\[ P(\tau,\xi,\eta,\zeta) = P_0\,F_t(\tau-\zeta)\,F_s(\xi,\eta,\zeta), \tag{F.3}\]

using the temporal envelope

\[ F_t = \exp\!\left[-\frac{(t-t_0-z/c)^2}{2\sigma_t^2}\right], \tag{F.4}\]

and the transverse profile

\[ F_s = \frac{w_{0x}}{w_x(z)}\frac{w_{0y}}{w_y(z)} \exp\!\left[-2\frac{x^2}{w_x^2(z)} - 2\frac{y^2}{w_y^2(z)}\right], \tag{F.5}\]

with

\[ w_{x,y}(z) = w_{0x,y}\sqrt{1+\frac{z^2}{z_{R,x,y}^2}}, \qquad z_{R,x,y} = \frac{\pi w_{0x,y}^2}{\lambda_L}. \tag{F.6}\]

The local photon number density is then

\[ n_\gamma(\mathbf x,t) = \frac{P(\mathbf x,t)}{\hbar\omega_L c}, \qquad \omega_L = \frac{2\pi c}{\lambda_L}. \tag{F.7}\]

F.9.3 Laser–Electron Inverse Compton Scattering

In the incident-particle rest frame, Thomson scattering is elastic,

\[ E_f' = E_i', \tag{F.8}\]

while Compton scattering gives

\[ E_f' = \frac{E_i'}{1 + \epsilon'(1-\cos\theta')}, \qquad \epsilon' = \frac{E_i'}{mc^2}. \tag{F.9}\]

For a head-on collision, the familiar Compton edge is

\[ E_{\gamma,\max} \simeq \frac{4\gamma^2 E_L}{1 + 4\gamma E_L/(mc^2)}, \tag{F.10}\]

which reduces in the Thomson limit to

\[ E_{\gamma,\max} \simeq 4\gamma^2 E_L. \tag{F.11}\]

The appendix also records the small-angle approximation

\[ E_\gamma(\theta) \simeq \frac{4\gamma^2 E_L}{1 + 4\gamma E_L/(mc^2) + \gamma^2\theta^2}. \tag{F.12}\]

The differential cross sections used in the design are:

\[ \frac{d\sigma_T}{d\Omega} = \frac{r_e^2}{2}(1+\cos^2\theta), \qquad \sigma_T = \frac{8\pi}{3}r_e^2, \tag{F.13}\]

and the unpolarized Klein–Nishina form

\[ \frac{d\sigma_{KN}}{d\Omega} = \frac{r_e^2}{2} \left(\frac{\omega_f'}{\omega_i'}\right)^2 \left[ \frac{\omega_f'}{\omega_i'} + \frac{\omega_i'}{\omega_f'} - \sin^2\theta' \right]. \tag{F.14}\]

The interaction probability is evaluated in the rest frame:

\[ P_{\mathrm{int}} = 1 - \exp\!\left(-n_\gamma'\,\sigma_{\mathrm{tot}}\,c\,\Delta t'\right). \tag{F.15}\]

F.9.4 gamma-gamma Luminosity

For two photons with energies E_1, E_2 and crossing angle psi,

\[ s = 2 E_1 E_2 (1-\cos\psi) \tag{F.16}\]

in natural units. The local luminosity density is

\[ \mathcal L(\mathbf x,t) = n_1(\mathbf x,t) n_2(\mathbf x,t) c (1-\cos\psi). \tag{F.17}\]

On a collision mesh, the exact pairwise histogram estimator for a cell of volume V_c and time step \Delta t is

\[ \Delta L_c(W_b) = \sum_{i\in c}\sum_{j\in c} \frac{w_i w_j}{V_c} \,c(1-\cos\psi_{ij})\,\Delta t\,\mathbf 1_{W_{ij}\in b}. \tag{F.18}\]

The helicity-resolved luminosity bins are accumulated with

\[ L^{J_z=0} \propto \tfrac12 L(1+\lambda_1\lambda_2), \qquad L^{J_z=2} \propto \tfrac12 L(1-\lambda_1\lambda_2). \tag{F.19}\]

F.9.5 Breit–Wheeler Pair Creation

The threshold condition is

\[ s \ge 4m_e^2 c^4, \tag{F.20}\]

with

\[ \beta = \sqrt{1 - \frac{4m_e^2 c^4}{s}}. \tag{F.21}\]

The total unpolarized cross section is

\[ \sigma_{BW}(s) = \frac{\pi r_e^2}{2}(1-\beta^2) \left[(3-\beta^4)\ln\frac{1+\beta}{1-\beta} - 2\beta(2-\beta^2)\right]. \tag{F.22}\]

The expected event weight for a pair of photon macro-particles in one cell is

\[ \mu_{ij} = \frac{w_i w_j}{V_c}\,\sigma_{BW}(s_{ij})\,c(1-\cos\psi_{ij})\,\Delta t. \tag{F.23}\]