def run_simulation(gamma_boost, use_separate_electron_species):
"""
Run a simulation with a laser pulse going through a gas jet of ionizable
N5+ atoms, and check the fraction of atoms that are in the N5+ state.
Parameters
----------
gamma_boost: float
The Lorentz factor of the frame in which the simulation is carried out.
use_separate_electron_species: bool
Whether to use separate electron species for each level, or
a single electron species for all levels.
"""
# The simulation box
zmax_lab = 20.e-6 # Length of the box along z (meters)
zmin_lab = 0.e-6
Nr = 3 # Number of gridpoints along r
rmax = 10.e-6 # Length of the box along r (meters)
Nm = 2 # Number of modes used
# The particles of the plasma
p_zmin = 5.e-6 # Position of the beginning of the plasma (meters)
p_zmax = 15.e-6
p_rmin = 0. # Minimal radial position of the plasma (meters)
p_rmax = 100.e-6 # Maximal radial position of the plasma (meters)
n_atoms = 0.2 # The atomic density is chosen very low,
# to avoid collective effects
p_nz = 2 # Number of particles per cell along z
p_nr = 1 # Number of particles per cell along r
p_nt = 4 # Number of particles per cell along theta
# Boosted frame
boost = BoostConverter(gamma_boost)
# Boost the different quantities
beta_boost = np.sqrt(1. - 1. / gamma_boost**2)
zmin, zmax = boost.static_length([zmin_lab, zmax_lab])
p_zmin, p_zmax = boost.static_length([p_zmin, p_zmax])
n_atoms, = boost.static_density([n_atoms])
# Increase the number of particles per cell in order to keep sufficient
# statistics for the evaluation of the ionization fraction
if gamma_boost > 1:
p_nz = int(2 * gamma_boost * (1 + beta_boost) * p_nz)
# The laser
a0 = 1.8 # Laser amplitude
lambda0_lab = 0.8e-6 # Laser wavelength
# Boost the laser wavelength before calculating the laser amplitude
lambda0, = boost.copropag_length([lambda0_lab], beta_object=1.)
# Duration and initial position of the laser
ctau = 10. * lambda0
z0 = -2 * ctau
# Calculate laser amplitude
omega = 2 * np.pi * c / lambda0
E0 = a0 * m_e * c * omega / e
B0 = E0 / c
def laser_func(F, x, y, z, t, amplitude, length_scale):
"""
Function that describes a Gaussian laser with infinite waist
"""
return( F + amplitude * math.cos( 2*np.pi*(z-c*t)/lambda0 ) * \
math.exp( - (z - c*t - z0)**2/ctau**2 ) )
# Resolution and number of timesteps
dz = lambda0 / 16.
dt = dz / c
Nz = int((zmax - zmin) / dz) + 1
N_step = int(
(2. * 40. * lambda0 + zmax - zmin) / (dz * (1 + beta_boost))) + 1
# Get the speed of the plasma
uz_m, = boost.longitudinal_momentum([0.])
v_plasma, = boost.velocity([0.])
# The diagnostics
diag_period = N_step - 1 # Period of the diagnostics in number of timesteps
# Initial ionization level of the Nitrogen atoms
level_start = 2
# Initialize the simulation object, with the neutralizing electrons
# No particles are created because we do not pass the density
sim = Simulation(Nz,
zmax,
Nr,
rmax,
Nm,
dt,
zmin=zmin,
v_comoving=v_plasma,
use_galilean=False,
boundaries='open',
use_cuda=use_cuda)
# Add the charge-neutralizing electrons
elec = sim.add_new_species(q=-e,
m=m_e,
n=level_start * n_atoms,
p_nz=p_nz,
p_nr=p_nr,
p_nt=p_nt,
p_zmin=p_zmin,
p_zmax=p_zmax,
p_rmin=p_rmin,
p_rmax=p_rmax,
continuous_injection=False,
uz_m=uz_m)
# Add the N atoms
ions = sim.add_new_species(q=0,
m=14. * m_p,
n=n_atoms,
p_nz=p_nz,
p_nr=p_nr,
p_nt=p_nt,
p_zmin=p_zmin,
p_zmax=p_zmax,
p_rmin=p_rmin,
p_rmax=p_rmax,
continuous_injection=False,
uz_m=uz_m)
# Add the target electrons
if use_separate_electron_species:
# Use a dictionary of electron species: one per ionizable level
target_species = {}
level_max = 6 # N can go up to N7+, but here we stop at N6+
for i_level in range(level_start, level_max):
target_species[i_level] = sim.add_new_species(q=-e, m=m_e)
else:
# Use the pre-existing, charge-neutralizing electrons
target_species = elec
level_max = None # Default is going up to N7+
# Define ionization
ions.make_ionizable(element='N',
level_start=level_start,
level_max=level_max,
target_species=target_species)
# Set the moving window
sim.set_moving_window(v=v_plasma)
# Add a laser to the fields of the simulation (external fields)
sim.external_fields = [
ExternalField(laser_func, 'Ex', E0, 0.),
ExternalField(laser_func, 'By', B0, 0.)
]
# Add a particle diagnostic
sim.diags = [
ParticleDiagnostic(
diag_period,
{"ions": ions},
particle_data=["position", "gamma", "weighting", "E", "B"],
# Test output of fields and gamma for standard
# (non-boosted) particle diagnostics
write_dir='tests/diags',
comm=sim.comm)
]
if gamma_boost > 1:
T_sim_lab = (2. * 40. * lambda0_lab + zmax_lab - zmin_lab) / c
sim.diags.append(
BackTransformedParticleDiagnostic(zmin_lab,
zmax_lab,
v_lab=0.,
dt_snapshots_lab=T_sim_lab / 2.,
Ntot_snapshots_lab=3,
gamma_boost=gamma_boost,
period=diag_period,
fldobject=sim.fld,
species={"ions": ions},
comm=sim.comm,
write_dir='tests/lab_diags'))
# Run the simulation
sim.step(N_step, use_true_rho=True)
# Check the fraction of N5+ ions at the end of the simulation
w = ions.w
ioniz_level = ions.ionizer.ionization_level
# Get the total number of N atoms/ions (all ionization levels together)
ntot = w.sum()
# Get the total number of N5+ ions
n_N5 = w[ioniz_level == 5].sum()
# Get the fraction of N5+ ions, and check that it is close to 0.32
N5_fraction = n_N5 / ntot
print('N5+ fraction: %.4f' % N5_fraction)
assert ((N5_fraction > 0.30) and (N5_fraction < 0.34))
# When different electron species are created, check the fraction of
# each electron species
if use_separate_electron_species:
for i_level in range(level_start, level_max):
n_N = w[ioniz_level == i_level].sum()
assert np.allclose(target_species[i_level].w.sum(), n_N)
# Check consistency in the regular openPMD diagnostics
ts = OpenPMDTimeSeries('./tests/diags/hdf5/')
last_iteration = ts.iterations[-1]
w, q = ts.get_particle(['w', 'charge'],
species="ions",
iteration=last_iteration)
# Check that the openPMD file contains the same number of N5+ ions
n_N5_openpmd = np.sum(w[(4.5 * e < q) & (q < 5.5 * e)])
assert np.isclose(n_N5_openpmd, n_N5)
# Remove openPMD files
shutil.rmtree('./tests/diags/')
# Check consistency of the back-transformed openPMD diagnostics
if gamma_boost > 1.:
ts = OpenPMDTimeSeries('./tests/lab_diags/hdf5/')
last_iteration = ts.iterations[-1]
w, q = ts.get_particle(['w', 'charge'],
species="ions",
iteration=last_iteration)
# Check that the openPMD file contains the same number of N5+ ions
n_N5_openpmd = np.sum(w[(4.5 * e < q) & (q < 5.5 * e)])
assert np.isclose(n_N5_openpmd, n_N5)
# Remove openPMD files
shutil.rmtree('./tests/lab_diags/')
def test_external_laser_field(show=False):
"Function that is run by py.test, when doing `python setup.py test`"
# Initialize the simulation
sim = Simulation(Nz,
zmax,
Nr,
rmax,
Nm,
dt,
p_zmin,
p_zmax,
0,
p_rmax,
p_nz,
p_nr,
p_nt,
n,
initialize_ions=False,
zmin=zmin,
use_cuda=use_cuda,
boundaries='periodic')
# Add the external fields
sim.external_fields = [
ExternalField(laser_func, 'Ex', a0 * m_e * c**2 * k0 / e, lambda0),
ExternalField(laser_func, 'By', a0 * m_e * c * k0 / e, lambda0)
]
# Prepare the arrays for the time history of the pusher
Nptcl = sim.ptcl[0].Ntot
x = np.zeros((N_step, Nptcl))
y = np.zeros((N_step, Nptcl))
z = np.zeros((N_step, Nptcl))
ux = np.zeros((N_step, Nptcl))
uz = np.zeros((N_step, Nptcl))
uy = np.zeros((N_step, Nptcl))
# Prepare the particles with proper transverse and longitudinal momentum
sim.ptcl[0].ux = a0 * np.sin(k0 * sim.ptcl[0].z)
sim.ptcl[0].uz[:] = 0.5 * sim.ptcl[0].ux**2
# Push the particles over N_step and record the corresponding history
for i in range(N_step):
# Record the history
x[i, :] = sim.ptcl[0].x[:]
y[i, :] = sim.ptcl[0].y[:]
z[i, :] = sim.ptcl[0].z[:]
ux[i, :] = sim.ptcl[0].ux[:]
uy[i, :] = sim.ptcl[0].uy[:]
uz[i, :] = sim.ptcl[0].uz[:]
# Take a simulation step
sim.step(1)
# Compute the analytical solution
t = dt * np.arange(N_step)
# Conservation of ux
ux_analytical = np.zeros((N_step, Nptcl))
uz_analytical = np.zeros((N_step, Nptcl))
for i in range(N_step):
ux_analytical[i, :] = a0 * np.sin(k0 * (z[i, :] - c * t[i]))
uz_analytical[i, :] = 0.5 * ux_analytical[i, :]**2
# Show the results
if show:
import matplotlib.pyplot as plt
plt.figure(figsize=(10, 5))
plt.subplot(211)
plt.plot(t, ux_analytical, '--')
plt.plot(t, ux, 'o')
plt.xlabel('t')
plt.ylabel('ux')
plt.subplot(212)
plt.plot(t, uz_analytical, '--')
plt.plot(t, uz, 'o')
plt.xlabel('t')
plt.ylabel('uz')
plt.show()
else:
assert np.allclose(ux, ux_analytical, atol=5.e-2)
assert np.allclose(uz, uz_analytical, atol=5.e-2)
def run_external_laser_field_simulation(show, gamma_boost=None):
"""
Runs a simulation with a set of particles whose motion corresponds
to that of a particle that is initially at rest (in the lab frame)
before being reached by a plane wave (propagating to the right)
In the lab frame, the motion is given by
ux = a0 sin ( k0(z-ct) )
uz = ux^2 / 2 (from the conservation of gamma - uz)
In the boosted frame, the motion is given by
ux = a0 sin ( k0 gamma0 (1-beta0) (z-ct) )
uz = - gamma0 beta0 + gamma0 (1-beta0) ux^2 / 2
"""
# Time parameters
dt = lambda0 / c / 200 # 200 points per laser period
N_step = 400 # Two laser periods
# Initialize BoostConverter object
if gamma_boost is None:
boost = BoostConverter(gamma0=1.)
else:
boost = BoostConverter(gamma_boost)
# Reduce time resolution, for the case of a boosted simulation
if gamma_boost is not None:
dt = dt * (1. + boost.beta0) / boost.gamma0
# Initialize the simulation
sim = Simulation(Nz,
zmax,
Nr,
rmax,
Nm,
dt,
initialize_ions=False,
zmin=zmin,
use_cuda=use_cuda,
boundaries='periodic',
gamma_boost=gamma_boost)
# Add electrons
sim.ptcl = []
sim.add_new_species(-e,
m_e,
n=n,
p_rmax=p_rmax,
p_nz=p_nz,
p_nr=p_nr,
p_nt=p_nt)
# Add the external fields
sim.external_fields = [
ExternalField(laser_func,
'Ex',
a0 * m_e * c**2 * k0 / e,
lambda0,
gamma_boost=gamma_boost),
ExternalField(laser_func,
'By',
a0 * m_e * c * k0 / e,
lambda0,
gamma_boost=gamma_boost)
]
# Prepare the arrays for the time history of the pusher
Nptcl = sim.ptcl[0].Ntot
x = np.zeros((N_step, Nptcl))
y = np.zeros((N_step, Nptcl))
z = np.zeros((N_step, Nptcl))
ux = np.zeros((N_step, Nptcl))
uy = np.zeros((N_step, Nptcl))
uz = np.zeros((N_step, Nptcl))
# Prepare the particles with proper transverse and longitudinal momentum,
# at t=0 in the simulation frame
k0p = k0 * boost.gamma0 * (1. - boost.beta0)
sim.ptcl[0].ux = a0 * np.sin(k0p * sim.ptcl[0].z)
sim.ptcl[0].uz[:] = -boost.gamma0*boost.beta0 \
+ boost.gamma0*(1-boost.beta0)*0.5*sim.ptcl[0].ux**2
# Push the particles over N_step and record the corresponding history
for i in range(N_step):
# Record the history
x[i, :] = sim.ptcl[0].x[:]
y[i, :] = sim.ptcl[0].y[:]
z[i, :] = sim.ptcl[0].z[:]
ux[i, :] = sim.ptcl[0].ux[:]
uy[i, :] = sim.ptcl[0].uy[:]
uz[i, :] = sim.ptcl[0].uz[:]
# Take a simulation step
sim.step(1)
# Compute the analytical solution
t = sim.dt * np.arange(N_step)
# Conservation of ux
ux_analytical = np.zeros((N_step, Nptcl))
uz_analytical = np.zeros((N_step, Nptcl))
for i in range(N_step):
ux_analytical[i, :] = a0 * np.sin(k0p * (z[i, :] - c * t[i]))
uz_analytical[i,:] = -boost.gamma0*boost.beta0 \
+ boost.gamma0*(1-boost.beta0)*0.5*ux_analytical[i,:]**2
# Show the results
if show:
import matplotlib.pyplot as plt
plt.figure(figsize=(10, 5))
plt.subplot(211)
plt.plot(t, ux_analytical, '--')
plt.plot(t, ux, 'o')
plt.xlabel('t')
plt.ylabel('ux')
plt.subplot(212)
plt.plot(t, uz_analytical, '--')
plt.plot(t, uz, 'o')
plt.xlabel('t')
plt.ylabel('uz')
plt.show()
else:
assert np.allclose(ux, ux_analytical, atol=5.e-2)
assert np.allclose(uz, uz_analytical, atol=5.e-2)
def run_simulation(gamma_boost):
"""
Run a simulation with a laser pulse going through a gas jet of ionizable
N5+ atoms, and check the fraction of atoms that are in the N5+ state.
Parameters
----------
gamma_boost: float
The Lorentz factor of the frame in which the simulation is carried out.
"""
# The simulation box
zmax_lab = 20.e-6 # Length of the box along z (meters)
zmin_lab = 0.e-6
Nr = 3 # Number of gridpoints along r
rmax = 10.e-6 # Length of the box along r (meters)
Nm = 2 # Number of modes used
# The particles of the plasma
p_zmin = 5.e-6 # Position of the beginning of the plasma (meters)
p_zmax = 15.e-6
p_rmin = 0. # Minimal radial position of the plasma (meters)
p_rmax = 100.e-6 # Maximal radial position of the plasma (meters)
n_e = 1. # The plasma density is chosen very low,
# to avoid collective effects
p_nz = 2 # Number of particles per cell along z
p_nr = 1 # Number of particles per cell along r
p_nt = 4 # Number of particles per cell along theta
# Boosted frame
boost = BoostConverter(gamma_boost)
# Boost the different quantities
beta_boost = np.sqrt(1. - 1. / gamma_boost**2)
zmin, zmax = boost.static_length([zmin_lab, zmax_lab])
p_zmin, p_zmax = boost.static_length([p_zmin, p_zmax])
n_e, = boost.static_density([n_e])
# Increase the number of particles per cell in order to keep sufficient
# statistics for the evaluation of the ionization fraction
if gamma_boost > 1:
p_nz = int(2 * gamma_boost * (1 + beta_boost) * p_nz)
# The laser
a0 = 1.8 # Laser amplitude
lambda0_lab = 0.8e-6 # Laser wavelength
# Boost the laser wavelength before calculating the laser amplitude
lambda0, = boost.copropag_length([lambda0_lab], beta_object=1.)
# Duration and initial position of the laser
ctau = 10. * lambda0
z0 = -2 * ctau
# Calculate laser amplitude
omega = 2 * np.pi * c / lambda0
E0 = a0 * m_e * c * omega / e
B0 = E0 / c
def laser_func(F, x, y, z, t, amplitude, length_scale):
"""
Function that describes a Gaussian laser with infinite waist
"""
return( F + amplitude * math.cos( 2*np.pi*(z-c*t)/lambda0 ) * \
math.exp( - (z - c*t - z0)**2/ctau**2 ) )
# Resolution and number of timesteps
dz = lambda0 / 16.
dt = dz / c
Nz = int((zmax - zmin) / dz) + 1
N_step = int(
(2. * 40. * lambda0 + zmax - zmin) / (dz * (1 + beta_boost))) + 1
# Get the speed of the plasma
uz_m, = boost.longitudinal_momentum([0.])
v_plasma, = boost.velocity([0.])
# The diagnostics
diag_period = N_step - 1 # Period of the diagnostics in number of timesteps
# Initialize the simulation object
sim = Simulation(
Nz,
zmax,
Nr,
rmax,
Nm,
dt,
p_zmax,
p_zmax, # No electrons get created because we pass p_zmin=p_zmax
p_rmin,
p_rmax,
p_nz,
p_nr,
p_nt,
n_e,
zmin=zmin,
initialize_ions=False,
v_comoving=v_plasma,
use_galilean=False,
boundaries='open',
use_cuda=use_cuda)
sim.set_moving_window(v=v_plasma)
# Add the N atoms
p_zmin, p_zmax, Npz = adapt_to_grid(sim.fld.interp[0].z, p_zmin, p_zmax,
p_nz)
p_rmin, p_rmax, Npr = adapt_to_grid(sim.fld.interp[0].r, p_rmin, p_rmax,
p_nr)
sim.ptcl.append(
Particles(q=e,
m=14. * m_p,
n=0.2 * n_e,
Npz=Npz,
zmin=p_zmin,
zmax=p_zmax,
Npr=Npr,
rmin=p_rmin,
rmax=p_rmax,
Nptheta=p_nt,
dt=dt,
use_cuda=use_cuda,
uz_m=uz_m,
grid_shape=sim.fld.interp[0].Ez.shape,
continuous_injection=False))
sim.ptcl[1].make_ionizable(element='N',
level_start=0,
target_species=sim.ptcl[0])
# Add a laser to the fields of the simulation (external fields)
sim.external_fields = [
ExternalField(laser_func, 'Ex', E0, 0.),
ExternalField(laser_func, 'By', B0, 0.)
]
# Add a field diagnostic
sim.diags = [
ParticleDiagnostic(diag_period, {"ions": sim.ptcl[1]},
write_dir='tests/diags',
comm=sim.comm)
]
if gamma_boost > 1:
T_sim_lab = (2. * 40. * lambda0_lab + zmax_lab - zmin_lab) / c
sim.diags.append(
BoostedParticleDiagnostic(zmin_lab,
zmax_lab,
v_lab=0.,
dt_snapshots_lab=T_sim_lab / 2.,
Ntot_snapshots_lab=3,
gamma_boost=gamma_boost,
period=diag_period,
fldobject=sim.fld,
species={"ions": sim.ptcl[1]},
comm=sim.comm,
write_dir='tests/lab_diags'))
# Run the simulation
sim.step(N_step, use_true_rho=True)
# Check the fraction of N5+ ions at the end of the simulation
w = sim.ptcl[1].w
ioniz_level = sim.ptcl[1].ionizer.ionization_level
# Get the total number of N atoms/ions (all ionization levels together)
ntot = w.sum()
# Get the total number of N5+ ions
n_N5 = w[ioniz_level == 5].sum()
# Get the fraction of N5+ ions, and check that it is close to 0.32
N5_fraction = n_N5 / ntot
print('N5+ fraction: %.4f' % N5_fraction)
assert ((N5_fraction > 0.30) and (N5_fraction < 0.34))
# Check consistency in the regular openPMD diagnostics
ts = OpenPMDTimeSeries('./tests/diags/hdf5/')
last_iteration = ts.iterations[-1]
w, q = ts.get_particle(['w', 'charge'],
species="ions",
iteration=last_iteration)
# Check that the openPMD file contains the same number of N5+ ions
n_N5_openpmd = np.sum(w[(4.5 * e < q) & (q < 5.5 * e)])
assert np.isclose(n_N5_openpmd, n_N5)
# Remove openPMD files
shutil.rmtree('./tests/diags/')
# Check consistency of the back-transformed openPMD diagnostics
if gamma_boost > 1.:
ts = OpenPMDTimeSeries('./tests/lab_diags/hdf5/')
last_iteration = ts.iterations[-1]
w, q = ts.get_particle(['w', 'charge'],
species="ions",
iteration=last_iteration)
# Check that the openPMD file contains the same number of N5+ ions
n_N5_openpmd = np.sum(w[(4.5 * e < q) & (q < 5.5 * e)])
assert np.isclose(n_N5_openpmd, n_N5)
# Remove openPMD files
shutil.rmtree('./tests/lab_diags/')
