def simulate_beam_focusing( z_injection_plane, write_dir ):
"""
Simulate a focusing beam in the boosted frame
Parameters
----------
z_injection_plane: float or None
when this is not None, the injection through a plane is
activated.
write_dir: string
The directory where the boosted diagnostics are written.
"""
# Initialize the simulation object
sim = Simulation( Nz, zmax, Nr, rmax, Nm, dt, zmin=zmin,
gamma_boost=gamma_boost, boundaries='open',
use_cuda=use_cuda, v_comoving=v_comoving )
# Note: no macroparticles get created because we do not pass
# the density and number of particle per cell
# Remove the plasma particles
sim.ptcl = []
# Initialize the bunch, along with its space charge
add_elec_bunch_gaussian( sim, sigma_r, sigma_z, n_emit, gamma0,
sigma_gamma, Q, N, tf=(z_focus-z0)/c, zf=z_focus, boost=boost,
z_injection_plane=z_injection_plane )
# Configure the moving window
sim.set_moving_window( v=c )
# Add a field diagnostic
sim.diags = [
BackTransformedParticleDiagnostic( zmin, zmax, c, dt_snapshot_lab,
Ntot_snapshot_lab, gamma_boost, period=100, fldobject=sim.fld,
species={'bunch':sim.ptcl[0]}, comm=sim.comm, write_dir=write_dir)
]
# Run the simulation
sim.step( N_step )
def run_and_check_laser_antenna(gamma_b,
show,
write_files,
z0,
v=0,
forward_propagating=True):
"""
Generic function, which runs and check the laser antenna for
both boosted frame and lab frame
Parameters
----------
gamma_b: float or None
The Lorentz factor of the boosted frame
show: bool
Whether to show the images of the laser as pop-up windows
write_files: bool
Whether to output openPMD data of the laser
v: float (m/s)
Speed of the laser antenna
"""
# Initialize the simulation object
sim = Simulation(Nz,
zmax,
Nr,
rmax,
Nm,
dt,
p_zmin=0,
p_zmax=0,
p_rmin=0,
p_rmax=0,
p_nz=2,
p_nr=2,
p_nt=2,
n_e=0.,
zmin=zmin,
use_cuda=use_cuda,
boundaries='open',
gamma_boost=gamma_b)
# Remove the particles
sim.ptcl = []
# Add the laser
add_laser(sim,
a0,
w0,
ctau,
z0,
zf=zf,
method='antenna',
z0_antenna=z0_antenna,
v_antenna=v,
gamma_boost=gamma_b,
fw_propagating=forward_propagating)
# Calculate the number of steps between each output
N_step = int(round(Ntot_step / N_show))
# Add diagnostic
if write_files:
sim.diags = [
FieldDiagnostic(N_step,
sim.fld,
comm=None,
fieldtypes=["rho", "E", "B", "J"])
]
# Loop over the iterations
print('Running the simulation...')
for it in range(N_show):
print('Diagnostic point %d/%d' % (it, N_show))
# Advance the Maxwell equations
sim.step(N_step, show_progress=False)
# Plot the fields during the simulation
if show == True:
show_fields(sim.fld.interp[1], 'Er')
# Finish the remaining iterations
sim.step(Ntot_step - N_show * N_step, show_progress=False)
# Check the transverse E and B field
Nz_half = int(sim.fld.interp[1].Nz / 2) + 2
z = sim.fld.interp[1].z[Nz_half:-(sim.comm.n_guard+sim.comm.n_damp+\
sim.comm.n_inject)]
r = sim.fld.interp[1].r
# Loop through the different fields
for fieldtype, info_in_real_part, factor in [ ('Er', True, 2.), \
('Et', False, 2.), ('Br', False, 2.*c), ('Bt', True, 2.*c) ]:
# factor correspond to the factor that has to be applied
# in order to get a value which is comparable to an electric field
# (Because of the definition of the interpolation grid, the )
field = getattr(sim.fld.interp[1], fieldtype)\
[Nz_half:-(sim.comm.n_guard+sim.comm.n_damp+\
sim.comm.n_inject)]
print('Checking %s' % fieldtype)
check_fields(factor * field, z, r, info_in_real_part, z0, gamma_b,
forward_propagating)
print('OK')
p_rmax,
p_nz,
p_nr,
p_nt,
n_e,
n_order=n_order,
v_comoving=-0.999999 * c,
use_galilean=True)
# Configure the moving window
#sim.moving_win = MovingWindow( sim.fld.interp[0],
# ncells_damp=2,
# ncells_zero=2 )
# Suppress the particles that were intialized by default and add the bunch
sim.ptcl = []
add_elec_bunch(sim,
gamma0,
n_e,
p_zmin,
p_zmax,
p_rmin,
p_rmax,
direction='backward')
# Show the initial fields
plt.figure(0)
sim.fld.interp[0].show('Ez')
plt.figure(1)
sim.fld.interp[0].show('Er')
plt.show()
def run_cpu_gpu_deposition(show=False, particle_shape='cubic'):
# Skip this test if cuda is not installed
if not cuda_installed:
return
# Perform deposition for a few timesteps, with both the CPU and GPU
for hardware in ['cpu', 'gpu']:
if hardware == 'cpu':
use_cuda = False
elif hardware == 'gpu':
use_cuda = True
# Initialize the simulation object
sim = Simulation(Nz,
zmax,
Nr,
rmax,
Nm,
dt,
zmin=zmin,
use_cuda=use_cuda,
particle_shape=particle_shape)
sim.ptcl = []
# Add an electron bunch (set the random seed first)
np.random.seed(0)
add_elec_bunch_gaussian(sim, sig_r, sig_z, n_emit, gamma0, sig_gamma,
Q, N)
# Add a field diagnostic
sim.diags = [
FieldDiagnostic(diag_period,
sim.fld,
fieldtypes=['rho', 'J'],
comm=sim.comm,
write_dir=os.path.join('tests', hardware))
]
### Run the simulation
sim.step(N_step)
# Check that the results are identical
ts_cpu = OpenPMDTimeSeries('tests/cpu/hdf5')
ts_gpu = OpenPMDTimeSeries('tests/gpu/hdf5')
for iteration in ts_cpu.iterations:
for field, coord in [('rho', ''), ('J', 'x'), ('J', 'z')]:
# Jy is not tested because it is zero
print('Testing %s at iteration %d' % (field + coord, iteration))
F_cpu, info = ts_cpu.get_field(field, coord, iteration=iteration)
F_gpu, info = ts_gpu.get_field(field, coord, iteration=iteration)
tolerance = 1.e-13 * (abs(F_cpu).max() + abs(F_gpu).max())
if not show:
assert np.allclose(F_cpu, F_gpu, atol=tolerance)
else:
if not np.allclose(F_cpu, F_gpu, atol=tolerance):
plot_difference(field, coord, iteration, F_cpu, F_gpu,
info)
# Remove the files used
shutil.rmtree('tests/cpu')
shutil.rmtree('tests/gpu')
def run_simulation(gamma_boost, show):
"""
Run a simulation with a relativistic electron bunch crosses a laser
Parameters
----------
gamma_boost: float
The Lorentz factor of the frame in which the simulation is carried out.
show: bool
Whether to show a plot of the angular distribution
"""
# Boosted frame
boost = BoostConverter(gamma_boost)
# The simulation timestep
diag_period = 100
N_step = 101 # Number of iterations to perform
# Calculate timestep to resolve the interaction with enough points
laser_duration_boosted, = boost.copropag_length([laser_duration],
beta_object=-1)
bunch_sigma_z_boosted, = boost.copropag_length([bunch_sigma_z],
beta_object=1)
dt = (4 * laser_duration_boosted + bunch_sigma_z_boosted / c) / N_step
# Initialize the simulation object
zmax, zmin = boost.copropag_length([zmax_lab, zmin_lab], beta_object=1.)
sim = Simulation(Nz,
zmax,
Nr,
rmax,
Nm,
dt,
p_zmin=0,
p_zmax=0,
p_rmin=0,
p_rmax=0,
p_nz=1,
p_nr=1,
p_nt=1,
n_e=1,
dens_func=None,
zmin=zmin,
boundaries='periodic',
use_cuda=use_cuda)
# Remove particles that were previously created
sim.ptcl = []
print('Initialized simulation')
# Add electron bunch (automatically converted to boosted-frame)
add_elec_bunch_gaussian(sim,
sig_r=1.e-6,
sig_z=bunch_sigma_z,
n_emit=0.,
gamma0=gamma_bunch_mean,
sig_gamma=gamma_bunch_rms,
Q=Q_bunch,
N=N_bunch,
tf=0.0,
zf=0.5 * (zmax + zmin),
boost=boost)
elec = sim.ptcl[0]
print('Initialized electron bunch')
# Add a photon species
photons = Particles(q=0,
m=0,
n=0,
Npz=1,
zmin=0,
zmax=0,
Npr=1,
rmin=0,
rmax=0,
Nptheta=1,
dt=sim.dt,
ux_m=0.,
uy_m=0.,
uz_m=0.,
ux_th=0.,
uy_th=0.,
uz_th=0.,
dens_func=None,
continuous_injection=False,
grid_shape=sim.fld.interp[0].Ez.shape,
particle_shape='linear',
use_cuda=sim.use_cuda)
sim.ptcl.append(photons)
print('Initialized photons')
# Activate Compton scattering for electrons of the bunch
elec.activate_compton(target_species=photons,
laser_energy=laser_energy,
laser_wavelength=laser_wavelength,
laser_waist=laser_waist,
laser_ctau=laser_ctau,
laser_initial_z0=laser_initial_z0,
ratio_w_electron_photon=50,
boost=boost)
print('Activated Compton')
# Add diagnostics
if write_hdf5:
sim.diags = [
ParticleDiagnostic(diag_period,
species={
'electrons': elec,
'photons': photons
},
comm=sim.comm)
]
# Get initial total momentum
initial_total_elec_px = (elec.w * elec.ux).sum() * m_e * c
initial_total_elec_py = (elec.w * elec.uy).sum() * m_e * c
initial_total_elec_pz = (elec.w * elec.uz).sum() * m_e * c
### Run the simulation
for species in sim.ptcl:
species.send_particles_to_gpu()
for i_step in range(N_step):
for species in sim.ptcl:
species.halfpush_x()
elec.handle_elementary_processes(sim.time + 0.5 * sim.dt)
for species in sim.ptcl:
species.halfpush_x()
# Increment time and run diagnostics
sim.time += sim.dt
sim.iteration += 1
for diag in sim.diags:
diag.write(sim.iteration)
# Print fraction of photons produced
if i_step % 10 == 0:
for species in sim.ptcl:
species.receive_particles_from_gpu()
simulated_frac = photons.w.sum() / elec.w.sum()
for species in sim.ptcl:
species.send_particles_to_gpu()
print( 'Iteration %d: Photon fraction per electron = %f' \
%(i_step, simulated_frac) )
for species in sim.ptcl:
species.receive_particles_from_gpu()
# Check estimation of photon fraction
check_photon_fraction(simulated_frac)
# Check conservation of momentum (is only conserved )
if elec.compton_scatterer.ratio_w_electron_photon == 1:
check_momentum_conservation(gamma_boost, photons, elec,
initial_total_elec_px,
initial_total_elec_py,
initial_total_elec_pz)
# Transform the photon momenta back into the lab frame
photon_u = 1. / photons.inv_gamma
photon_lab_pz = boost.gamma0 * (photons.uz + boost.beta0 * photon_u)
photon_lab_p = boost.gamma0 * (photon_u + boost.beta0 * photons.uz)
# Plot the scaled angle and frequency
if show:
import matplotlib.pyplot as plt
# Bin the photons on a grid in frequency and angle
freq_min = 0.5
freq_max = 1.2
N_freq = 500
gammatheta_min = 0.
gammatheta_max = 1.
N_gammatheta = 100
hist_range = [[freq_min, freq_max], [gammatheta_min, gammatheta_max]]
extent = [freq_min, freq_max, gammatheta_min, gammatheta_max]
fundamental_frequency = 4 * gamma_bunch_mean**2 * c / laser_wavelength
photon_scaled_freq = photon_lab_p * c / (h * fundamental_frequency)
gamma_theta = gamma_bunch_mean * np.arccos(
photon_lab_pz / photon_lab_p)
grid, freq_bins, gammatheta_bins = np.histogram2d(
photon_scaled_freq,
gamma_theta,
weights=photons.w,
range=hist_range,
bins=[N_freq, N_gammatheta])
# Normalize by solid angle, frequency and number of photons
dw = (freq_bins[1] - freq_bins[0]) * 2 * np.pi * fundamental_frequency
dtheta = (gammatheta_bins[1] - gammatheta_bins[0]) / gamma_bunch_mean
domega = 2. * np.pi * np.sin(
gammatheta_bins / gamma_bunch_mean) * dtheta
grid /= dw * domega[np.newaxis, 1:] * elec.w.sum()
grid = np.where(grid == 0, np.nan, grid)
plt.imshow(grid.T,
origin='lower',
extent=extent,
cmap='gist_earth',
aspect='auto',
vmax=1.8e-16)
plt.title('Particles, $d^2N/d\omega \,d\Omega$')
plt.xlabel('Scaled energy ($\omega/4\gamma^2\omega_\ell$)')
plt.ylabel(r'$\gamma \theta$')
plt.colorbar()
# Plot theory
plt.plot(1. / (1 + gammatheta_bins**2), gammatheta_bins, color='r')
plt.show()
plt.clf()
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,
initialize_ions=False,
v_comoving=v_plasma,
use_galilean=False,
boundaries='open',
use_cuda=use_cuda)
sim.ptcl = []
# 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 propagate_pulse(Nz,
Nr,
Nm,
zmin,
zmax,
Lr,
L_prop,
zf,
dt,
N_diag,
w0,
ctau,
k0,
E0,
m,
N_show,
n_order,
rtol,
boundaries,
v_window=0,
use_galilean=False,
v_comoving=0,
show=False):
"""
Propagate the beam over a distance L_prop in Nt steps,
and extracts the waist and a0 at each step.
Parameters
----------
show : bool
Wether to show the fields, so that the user can manually check
the agreement with the theory.
If True, this will periodically show the map of the fields (with
a period N_show), as well as (eventually) the evoluation of a0 and w0.
If False, this
N_diag : int
Number of diagnostic points (i.e. measure of waist and a0)
along the propagation
Nz, Nr : int
The number of points on the grid in z and r respectively
Nm : int
The number of modes in the azimuthal direction
zmin, zmax : float
The limits of the box in z
Lr : float
The size of the box in the r direction
(In the case of Lr, this is the distance from the *axis*
to the outer boundary)
L_prop : float
The total propagation distance (in meters)
zf : float
The position of the focal plane of the laser (only works for m=1)
dt : float
The timestep of the simulation
w0 : float
The initial waist of the laser (in meters)
ctau : float
The initial temporal waist of the laser (in meters)
k0 : flat
The central wavevector of the laser (in meters^-1)
E0 : float
The initial E0 of the pulse
m : int
Index of the mode to be tested
For m = 1 : test with a gaussian, linearly polarized beam
For m = 0 : test with an annular beam, polarized in E_theta
n_order : int
Order of the stencil
rtol : float
Relative precision with which the results are tested
boundaries : string
Type of boundary condition
Either 'open' or 'periodic'
v_window : float
Speed of the moving window
v_comoving : float
Velocity at which the currents are assumed to move
use_galilean: bool
Whether to use a galilean frame that moves at the speed v_comoving
Returns
-------
A dictionary containing :
- 'E' : 1d array containing the values of the amplitude
- 'w' : 1d array containing the values of waist
- 'fld' : the Fields object at the end of the simulation.
"""
# Initialize the simulation object
sim = Simulation(Nz,
zmax,
Nr,
Lr,
Nm,
dt,
p_zmin=0,
p_zmax=0,
p_rmin=0,
p_rmax=0,
p_nz=2,
p_nr=2,
p_nt=2,
n_e=0.,
n_order=n_order,
zmin=zmin,
use_cuda=use_cuda,
boundaries=boundaries,
v_comoving=v_comoving,
exchange_period=1,
use_galilean=use_galilean)
# Remove the particles
sim.ptcl = []
# Set the moving window object
if v_window != 0:
sim.set_moving_window(v=v_window)
# Initialize the laser fields
z0 = (zmax + zmin) / 2
init_fields(sim, w0, ctau, k0, z0, zf, E0, m)
# Create the arrays to get the waist and amplitude
w = np.zeros(N_diag)
E = np.zeros(N_diag)
# Calculate the number of steps to run between each diagnostic
Ntot_step = int(round(L_prop / (c * dt)))
N_step = int(round(Ntot_step / N_diag))
# Loop over the iterations
print('Running the simulation...')
for it in range(N_diag):
print('Diagnostic point %d/%d' % (it, N_diag))
# Fit the fields to find the waist and a0
w[it], E[it] = fit_fields(sim.fld, m)
# Plot the fields during the simulation
if show == True and it % N_show == 0:
import matplotlib.pyplot as plt
plt.clf()
sim.fld.interp[m].show('Er')
plt.show()
# Advance the Maxwell equations
sim.step(N_step, show_progress=False)
# Get the analytical solution
z_prop = c * dt * N_step * np.arange(N_diag)
ZR = 0.5 * k0 * w0**2
w_analytic = w0 * np.sqrt(1 + (z_prop - zf)**2 / ZR**2)
E_analytic = E0 / (1 + (z_prop - zf)**2 / ZR**2)**(1. / 2)
# Either plot the results and check them manually
if show is True:
import matplotlib.pyplot as plt
plt.suptitle('Diffraction of a pulse in the mode %d' % m)
plt.subplot(121)
plt.plot(1.e6 * z_prop, 1.e6 * w, 'o', label='Simulation')
plt.plot(1.e6 * z_prop, 1.e6 * w_analytic, '--', label='Theory')
plt.xlabel('z (microns)')
plt.ylabel('w (microns)')
plt.title('Waist')
plt.legend(loc=0)
plt.subplot(122)
plt.plot(1.e6 * z_prop, E, 'o', label='Simulation')
plt.plot(1.e6 * z_prop, E_analytic, '--', label='Theory')
plt.xlabel('z (microns)')
plt.ylabel('E')
plt.legend(loc=0)
plt.title('Amplitude')
plt.show()
# or automatically check that the theoretical and simulated curves
# of w and E are close
else:
assert np.allclose(w, w_analytic, rtol=rtol)
assert np.allclose(E, E_analytic, rtol=5.e-3)
print('The simulation results agree with the theory to %e.' % rtol)
# Return a dictionary of the results
return ({'E': E, 'w': w, 'fld': sim.fld})
def test_boosted_output(gamma_boost=10.):
"""
# TODO
Parameters
----------
gamma_boost: float
The Lorentz factor of the frame in which the simulation is carried out.
"""
# The simulation box
Nz = 500 # Number of gridpoints along z
zmax_lab = 0.e-6 # Length of the box along z (meters)
zmin_lab = -20.e-6
Nr = 10 # Number of gridpoints along r
rmax = 10.e-6 # Length of the box along r (meters)
Nm = 2 # Number of modes used
# Number of timesteps
N_steps = 500
diag_period = 20 # Period of the diagnostics in number of timesteps
dt_lab = (zmax_lab - zmin_lab) / Nz * 1. / c
T_sim_lab = N_steps * dt_lab
# Move into directory `tests`
os.chdir('./tests')
# Initialize the simulation object
sim = Simulation(
Nz,
zmax_lab,
Nr,
rmax,
Nm,
dt_lab,
0,
0, # No electrons get created because we pass p_zmin=p_zmax=0
0,
rmax,
1,
1,
4,
n_e=0,
zmin=zmin_lab,
initialize_ions=False,
gamma_boost=gamma_boost,
v_comoving=-0.9999 * c,
boundaries='open',
use_cuda=use_cuda)
sim.set_moving_window(v=c)
# Remove the electron species
sim.ptcl = []
# Add a Gaussian electron bunch
# Note: the total charge is 0 so all fields should remain 0
# throughout the simulation. As a consequence, the motion of the beam
# is a mere translation.
N_particles = 3000
add_elec_bunch_gaussian(sim,
sig_r=1.e-6,
sig_z=1.e-6,
n_emit=0.,
gamma0=100,
sig_gamma=0.,
Q=0.,
N=N_particles,
zf=0.5 * (zmax_lab + zmin_lab),
boost=BoostConverter(gamma_boost))
sim.ptcl[0].track(sim.comm)
# openPMD diagnostics
sim.diags = [
BackTransformedParticleDiagnostic(zmin_lab,
zmax_lab,
v_lab=c,
dt_snapshots_lab=T_sim_lab / 3.,
Ntot_snapshots_lab=3,
gamma_boost=gamma_boost,
period=diag_period,
fldobject=sim.fld,
species={"bunch": sim.ptcl[0]},
comm=sim.comm)
]
# Run the simulation
sim.step(N_steps)
# Check consistency of the back-transformed openPMD diagnostics:
# Make sure that all the particles were retrived by checking particle IDs
ts = OpenPMDTimeSeries('./lab_diags/hdf5/')
ref_pid = np.sort(sim.ptcl[0].tracker.id)
for iteration in ts.iterations:
pid, = ts.get_particle(['id'], iteration=iteration)
pid = np.sort(pid)
assert len(pid) == N_particles
assert np.all(ref_pid == pid)
# Remove openPMD files
shutil.rmtree('./lab_diags/')
os.chdir('../')
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)
