First draft of spectrometer example.

examples/achromatic_spectrometer/README.rst

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+.. _examples-fodo:
+
+FODO Cell
+=========
+
+Stable FODO cell with a zero-current phase advance of 67.8 degrees.
+
+The matched Twiss parameters at entry are:
+
+* :math:`\beta_\mathrm{x} = 2.82161941` m
+* :math:`\alpha_\mathrm{x} = -1.59050035`
+* :math:`\beta_\mathrm{y} = 2.82161941` m
+* :math:`\alpha_\mathrm{y} = 1.59050035`
+
+We use a 2 GeV electron beam with initial unnormalized rms emittance of 2 nm.
+
+The second moments of the particle distribution after the FODO cell should coincide with the second moments of the particle distribution before the FODO cell, to within the level expected due to noise due to statistical sampling.
+
+In this test, the initial and final values of :math:`\sigma_x`, :math:`\sigma_y`, :math:`\sigma_t`, :math:`\epsilon_x`, :math:`\epsilon_y`, and :math:`\epsilon_t` must agree with nominal values.
+
+
+Run
+---
+
+This example can be run **either** as:
+
+* **Python** script: ``python3 run_fodo.py`` or
+* ImpactX **executable** using an input file: ``impactx input_fodo.in``
+
+For `MPI-parallel <https://www.mpi-forum.org>`__ runs, prefix these lines with ``mpiexec -n 4 ...`` or ``srun -n 4 ...``, depending on the system.
+
+.. tab-set::
+
+   .. tab-item:: Python: Script
+
+       .. literalinclude:: run_fodo.py
+          :language: python3
+          :caption: You can copy this file from ``examples/fodo/run_fodo.py``.
+
+   .. tab-item:: Executable: Input File
+
+       .. literalinclude:: input_fodo.in
+          :language: ini
+          :caption: You can copy this file from ``examples/fodo/input_fodo.in``.
+
+
+Analyze
+-------
+
+We run the following script to analyze correctness:
+
+.. dropdown:: Script ``analysis_fodo.py``
+
+   .. literalinclude:: analysis_fodo.py
+      :language: python3
+      :caption: You can copy this file from ``examples/fodo/analysis_fodo.py``.
+
+
+Visualize
+---------
+
+You can run the following script to visualize the beam evolution over time:
+
+.. dropdown:: Script ``plot_fodo.py``
+
+   .. literalinclude:: plot_fodo.py
+      :language: python3
+      :caption: You can copy this file from ``examples/fodo/plot_fodo.py``.
+
+.. figure:: https://user-images.githubusercontent.com/1353258/180287840-8561f6fd-278f-4856-abd8-04fbdb78c8ff.png
+   :alt: focusing, defocusing and preserved emittane in our FODO cell benchmark.
+
+   FODO transversal beam width and emittance evolution
+
+.. figure:: https://user-images.githubusercontent.com/1353258/180287845-eb0210a7-2500-4aa9-844c-67fb094329d3.png
+   :alt: focusing, defocusing and phase space rotation in our FODO cell benchmark.
+
+   FODO transversal beam width and phase space evolution

examples/achromatic_spectrometer/analysis_spectrometer.py

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+#!/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["momentum_x"], moment=2) ** 0.5
+    sigy = moment(beam["position_y"], moment=2) ** 0.5
+    sigpy = moment(beam["momentum_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 = 10000
+assert num_particles == len(initial)
+assert num_particles == len(final)
+
+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 = 2.2 * 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],
+    [
+        7.5451170454175073e-005,
+        7.5441588239210947e-005,
+        9.9775878164077539e-004,
+        1.9959540393751392e-009,
+        2.0175015289132990e-009,
+        2.0013820193294972e-006,
+    ],
+    rtol=rtol,
+    atol=atol,
+)
+
+
+print("")
+print("Final Beam:")
+sigx, sigy, sigt, emittance_x, emittance_y, emittance_t = get_moments(final)
+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 = 2.2 * 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],
+    [
+        7.4790118496224206e-005,
+        7.5357525169680140e-005,
+        9.9775879288128088e-004,
+        1.9959539836392703e-009,
+        2.0175014668882125e-009,
+        2.0013820380883801e-006,
+    ],
+    rtol=rtol,
+    atol=atol,
+)

examples/achromatic_spectrometer/input_spectrometer.in

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+###############################################################################
+# Particle Beam(s)
+###############################################################################
+beam.npart = 10000
+beam.units = static
+beam.kin_energy = 1.0e3
+beam.charge = 1.0e-9
+beam.particle = electron
+beam.distribution = waterbag
+beam.sigmaX = 3.162277660e-6
+beam.sigmaY = 3.162277660e-6
+beam.sigmaT = 1.0e-3
+beam.sigmaPx = 3.16227766017e-4
+beam.sigmaPy = 3.16227766017e-4
+beam.sigmaPt = 2.0e-2
+beam.muxpx = 0.0
+beam.muypy = 0.0
+beam.mutpt = 0.0
+
+
+###############################################################################
+# Beamline: lattice elements and segments
+###############################################################################
+lattice.elements = monitor bend1 plasma_lens drift1 monitor
+lattice.nslice = 25
+
+monitor.type = beam_monitor
+monitor.backend = h5
+
+bend1.type = sbend_exact
+bend1.ds = 1.0
+bend1.phi = 10.0
+bend1.B = 0.0
+
+plasma_lens.type = line
+plasma_lens.elements = plend dr pl dr pl dr pl dr pl dr p1 dr p1 dr p1 dr p1 dr p1 dr p1end
+
+p1end.type = tapered_plasma_lens
+p1end.k = 125.0
+p1end.taper = 11.488289081903567
+p1end.units = 0
+
+p1.type = tapered_plasma_lens
+p1.k = 250.0  #focal length 0.5 m
+p1.taper = 11.488289081903567
+p1.units = 0
+
+dr.type = drift
+dr.ds = 0.25
+
+drift1.type = drift
+drift1.ds = 1.0
+
+###############################################################################
+# Algorithms
+###############################################################################
+algo.particle_shape = 2
+algo.space_charge = false
+
+
+###############################################################################
+# Diagnostics
+###############################################################################
+diag.slice_step_diagnostics = true

examples/achromatic_spectrometer/run_spectrometer.py

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+#!/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 = 2.0e3  # reference energy
+bunch_charge_C = 1.0e-9  # used with space charge
+npart = 10000  # 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.Waterbag(
+    sigmaX=3.9984884770e-5,
+    sigmaY=3.9984884770e-5,
+    sigmaT=1.0e-3,
+    sigmaPx=2.6623538760e-5,
+    sigmaPy=2.6623538760e-5,
+    sigmaPt=2.0e-3,
+    muxpx=-0.846574929020762,
+    muypy=0.846574929020762,
+    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
+fodo = [
+    monitor,
+    elements.Drift(ds=0.25, nslice=ns),
+    monitor,
+    elements.Quad(ds=1.0, k=1.0, nslice=ns),
+    monitor,
+    elements.Drift(ds=0.5, nslice=ns),
+    monitor,
+    elements.Quad(ds=1.0, k=-1.0, nslice=ns),
+    monitor,
+    elements.Drift(ds=0.25, nslice=ns),
+    monitor,
+]
+# assign a fodo segment
+sim.lattice.extend(fodo)
+
+# run simulation
+sim.evolve()
+
+# clean shutdown
+sim.finalize()