Apochromatic Drift-Quadrupole Beamline

Electron beam matched to the 1st-order apochromatic drift-quadrupole beamline appearing in Fig. 4a of: C. A. Lindstrom and E. Adli, “Design of general apochromatic drift-quadrupole beam lines,” Phys. Rev. Accel. Beams 19, 071002 (2016).

The matched Twiss parameters at entry are:

\(\beta_\mathrm{x} = 0.325\) m
\(\alpha_\mathrm{x} = 0\)
\(\beta_\mathrm{y} = 0.325\) m
\(\alpha_\mathrm{y} = 0\)

We use a 100 GeV electron beam with an initially 6D Gaussian distribution of normalized rms emittance 1 micron and relative energy spread of 1%.

The second moments of the particle distribution after the focusing beamline should coincide with the second moments of the particle distribution before the beamline, to within the level expected due to noise due to statistical sampling. The emittance growth due to chromatic effects remain below 1%. In the absence of chromatic correction, the projected emittance growth is near 10%.

In this test, the initial and final values of \(\sigma_x\), \(\sigma_y\), \(\sigma_t\), \(\epsilon_x\), \(\epsilon_y\), and \(\epsilon_t\) must agree with nominal values.

Run

This example can be run either as:

Python script: python3 run_apochromatic.py or
ImpactX executable using an input file: impactx input_apochromatic.in

For MPI-parallel runs, prefix these lines with mpiexec -n 4 ... or srun -n 4 ..., depending on the system.

Python: Script

Listing 113 You can copy this file from examples/apochromatic/run_apochromatic.py.

#!/usr/bin/env python3
#
# Copyright 2022-2023 ImpactX contributors
# Authors: Axel Huebl, Chad Mitchell
# License: BSD-3-Clause-LBNL
#
# -*- coding: utf-8 -*-

from impactx import ImpactX, distribution, elements

sim = ImpactX()

# set numerical parameters and IO control
sim.particle_shape = 2  # B-spline order
sim.space_charge = False
# sim.diagnostics = False  # benchmarking
sim.slice_step_diagnostics = True

# domain decomposition & space charge mesh
sim.init_grids()

# load a 2 GeV electron beam with an initial
# unnormalized rms emittance of 2 nm
kin_energy_MeV = 100.0e3  # reference energy
bunch_charge_C = 1.0e-9  # used with space charge
npart = 100000  # number of macro particles

#   reference particle
ref = sim.particle_container().ref_particle()
ref.set_charge_qe(-1.0).set_mass_MeV(0.510998950).set_kin_energy_MeV(kin_energy_MeV)

#   particle bunch
distr = distribution.Gaussian(
    lambdaX=1.288697604e-6,
    lambdaY=1.288697604e-6,
    lambdaT=1.0e-6,
    lambdaPx=3.965223396e-6,
    lambdaPy=3.965223396e-6,
    lambdaPt=0.01,  # 1% energy spread
    muxpx=0.0,
    muypy=0.0,
    mutpt=0.0,
)
sim.add_particles(bunch_charge_C, distr, npart)

# add beam diagnostics
monitor = elements.BeamMonitor("monitor", backend="h5")

# design the accelerator lattice)
ns = 25  # number of slices per ds in the element

# Drift elements
dr1 = elements.ChrDrift(ds=1.0, nslice=ns)
dr2 = elements.ChrDrift(ds=10.0, nslice=ns)

# Quad elements
q1 = elements.ChrQuad(ds=1.2258333333, k=0.5884, nslice=ns)
q2 = elements.ChrQuad(ds=1.5677083333, k=-0.7525, nslice=ns)
q3 = elements.ChrQuad(ds=1.205625, k=0.5787, nslice=ns)
q4 = elements.ChrQuad(ds=1.2502083333, k=-0.6001, nslice=ns)
q5 = elements.ChrQuad(ds=1.2502083333, k=0.6001, nslice=ns)
q6 = elements.ChrQuad(ds=1.205625, k=-0.5787, nslice=ns)
q7 = elements.ChrQuad(ds=1.5677083333, k=0.7525, nslice=ns)
q8 = elements.ChrQuad(ds=1.2258333333, k=-0.5884, nslice=ns)

lattice_line = [monitor, dr1, q1, q2, q3, dr2, q4, q5, dr2, q6, q7, q8, dr1, monitor]

# define the lattice
sim.lattice.extend(lattice_line)

# run simulation
sim.evolve()

# clean shutdown
sim.finalize()

Executable: Input File

Listing 114 You can copy this file from examples/apochromatic/input_apochromatic.in.

###############################################################################
# Particle Beam(s)
###############################################################################
beam.npart = 100000
beam.units = static
beam.kin_energy = 100.0e3  # 100 GeV nominal energy
beam.charge = 1.0e-9
beam.particle = electron
beam.distribution = gaussian
beam.lambdaX = 1.288697604e-6
beam.lambdaY = 1.288697604e-6
beam.lambdaT = 1.0e-6
beam.lambdaPx = 3.965223396e-6
beam.lambdaPy = 3.965223396e-6
beam.lambdaPt = 0.01  #1% energy spread
beam.muxpx = 0.0
beam.muypy = 0.0
beam.mutpt = 0.0

###############################################################################
# Beamline: lattice elements and segments
###############################################################################
lattice.elements = monitor dr1 q1 q2 q3 dr2 q4 q5 dr2 q6 q7 q8 dr1 monitor
lattice.nslice = 1

monitor.type = beam_monitor
monitor.backend = h5

dr1.type = drift_chromatic
dr1.ds = 1.0

dr2.type = drift_chromatic
dr2.ds = 10.0

q1.type = quad_chromatic
q1.ds = 1.2258333333
q1.k = 0.5884

q2.type = quad_chromatic
q2.ds = 1.5677083333
q2.k = -0.7525

q3.type = quad_chromatic
q3.ds = 1.205625
q3.k = 0.5787

q4.type = quad_chromatic
q4.ds = 1.2502083333
q4.k = -0.6001

q5.type = quad_chromatic
q5.ds = 1.2502083333
q5.k = 0.6001

q6.type = quad_chromatic
q6.ds = 1.205625
q6.k = -0.5787

q7.type = quad_chromatic
q7.ds = 1.5677083333
q7.k = 0.7525

q8.type = quad_chromatic
q8.ds = 1.2258333333
q8.k = -0.5884

###############################################################################
# Algorithms
###############################################################################
algo.particle_shape = 2
algo.space_charge = false


###############################################################################
# Diagnostics
###############################################################################
diag.slice_step_diagnostics = true

Analyze

We run the following script to analyze correctness:

Script analysis_apochromatic.py

Listing 115 You can copy this file from examples/apochromatic/analysis_apochromatic.py.

#!/usr/bin/env python3
#
# Copyright 2022-2023 ImpactX contributors
# Authors: Axel Huebl, Chad Mitchell
# License: BSD-3-Clause-LBNL
#


import numpy as np
import openpmd_api as io
from scipy.stats import moment


def get_moments(beam):
    """Calculate standard deviations of beam position & momenta
    and emittance values

    Returns
    -------
    sigx, sigy, sigt, emittance_x, emittance_y, emittance_t
    """
    sigx = moment(beam["position_x"], moment=2) ** 0.5  # variance -> std dev.
    sigpx = moment(beam["divergence_x"], moment=2) ** 0.5
    sigy = moment(beam["position_y"], moment=2) ** 0.5
    sigpy = moment(beam["divergence_y"], moment=2) ** 0.5
    sigt = moment(beam["position_t"], moment=2) ** 0.5
    sigpt = moment(beam["momentum_t"], moment=2) ** 0.5

    epstrms = beam.cov(ddof=0)
    emittance_x = (sigx**2 * sigpx**2 - epstrms["position_x"]["momentum_x"] ** 2) ** 0.5
    emittance_y = (sigy**2 * sigpy**2 - epstrms["position_y"]["momentum_y"] ** 2) ** 0.5
    emittance_t = (sigt**2 * sigpt**2 - epstrms["position_t"]["momentum_t"] ** 2) ** 0.5

    return (sigx, sigy, sigt, emittance_x, emittance_y, emittance_t)


# initial/final beam
series = io.Series("diags/openPMD/monitor.h5", io.Access.read_only)
last_step = list(series.iterations)[-1]
initial = series.iterations[1].particles["beam"].to_df()
final = series.iterations[last_step].particles["beam"].to_df()

# compare number of particles
num_particles = 100000
assert num_particles == len(initial)
assert num_particles == len(final)

scale = (
    (1.0 - initial.momentum_t) ** 2
    + (initial.momentum_x) ** 2
    + (initial.momentum_y) ** 2
)
xp = initial.momentum_x / np.sqrt(scale)
initial["divergence_x"] = xp
yp = initial.momentum_y / np.sqrt(scale)
initial["divergence_y"] = yp

print("Initial Beam:")
sigx, sigy, sigt, emittance_x, emittance_y, emittance_t = get_moments(initial)
print(f"  sigx={sigx:e} sigy={sigy:e} sigt={sigt:e}")
print(
    f"  emittance_x={emittance_x:e} emittance_y={emittance_y:e} emittance_t={emittance_t:e}"
)

atol = 0.0  # ignored
rtol = 3.0 * num_particles**-0.5  # from random sampling of a smooth distribution
print(f"  rtol={rtol} (ignored: atol~={atol})")

assert np.allclose(
    [sigx, sigy, sigt, emittance_x, emittance_y, emittance_t],
    [
        1.288697604e-6,
        1.288697604e-6,
        1.0e-6,
        5.10997388810014764e-12,
        5.10997388810014764e-12,
        1.0e-8,
    ],
    rtol=rtol,
    atol=atol,
)


scale = (
    (1.0 - final.momentum_t) ** 2 + (final.momentum_x) ** 2 + (final.momentum_y) ** 2
)
xp = final.momentum_x / np.sqrt(scale)
final["divergence_x"] = xp
yp = final.momentum_y / np.sqrt(scale)
final["divergence_y"] = yp

print("")
print("Final Beam:")
sigx, sigy, sigt, emittance_xf, emittance_yf, emittance_tf = get_moments(final)
demittance_x = 100 * (emittance_xf - emittance_x) / emittance_x
demittance_y = 100 * (emittance_yf - emittance_y) / emittance_y
demittance_t = 100 * (emittance_tf - emittance_t) / emittance_t

print(f"  sigx={sigx:e} sigy={sigy:e} sigt={sigt:e}")
print(
    f"  emittance change x (%)={demittance_x:e} emittance change y (%)={demittance_y:e} emittance change t (%)={demittance_t:e}"
)

atol = 0.0  # ignored
rtol = 26.0 * num_particles**-0.5  # from random sampling of a smooth distribution
print(f"  rtol={rtol} (ignored: atol~={atol})")

assert np.allclose(
    [sigx, sigy, sigt, demittance_x, demittance_y, emittance_t],
    [
        1.245e-6,
        1.245e-6,
        1.0e-6,
        0.94,
        0.94,
        1.0e-8,
    ],
    rtol=rtol,
    atol=atol,
)

Apochromatic Drift-Plasma Lens Beamline

Electron beam matched to the 3rd-order apochromatic drift-plasma lens beamline appearing in Fig. 4b of: C. A. Lindstrom and E. Adli, “Design of general apochromatic drift-quadrupole beam lines,” Phys. Rev. Accel. Beams 19, 071002 (2016).