def check_fields(interp1_complex, z, r, info_in_real_part, z0, gamma_b, forward_propagating, show_difference=False): """ Check the real and imaginary part of the interpolation grid agree with the theory by: - Checking that the part (real or imaginary) that does not carry information is zero - Extracting the a0 from the other part and comparing it to the predicted value - Using the extracted value of a0 to compare the simulated profile with a gaussian profile """ # Extract the part that has information if info_in_real_part: interp1 = interp1_complex.real zero_part = interp1_complex.imag else: interp1 = interp1_complex.imag zero_part = interp1_complex.real # Control that the part that has no information is 0 assert np.allclose(0., zero_part, atol=1.e-6 * interp1.max()) # Get the predicted properties of the laser in the boosted frame if gamma_b is None: boost = BoostConverter(1.) else: boost = BoostConverter(gamma_b) ctau_b, lambda0_b, Lprop_b, z0_b = \ boost.copropag_length([ctau, 0.8e-6, Lprop, z0]) # Take into account whether the pulse is propagating forward or backward if not forward_propagating: Lprop_b = -Lprop_b # Fit the on-axis profile to extract a0 def fit_function(z, a0, z0_phase): return (gaussian_laser(z, r[0], a0, z0_phase, z0_b + Lprop_b, ctau_b, lambda0_b)) fit_result = curve_fit(fit_function, z, interp1[:, 0], p0=np.array([a0, z0_b + Lprop_b])) a0_fit, z0_fit = fit_result[0] # Check that the a0 agrees within 5% of the predicted value assert abs(abs(a0_fit) - a0) / a0 < 0.05 # Calculate predicted fields r2d, z2d = np.meshgrid(r, z) # Factor 0.5 due to the definition of the interpolation grid interp1_predicted = gaussian_laser(z2d, r2d, a0_fit, z0_fit, z0_b + Lprop_b, ctau_b, lambda0_b) # Plot the difference if show_difference: import matplotlib.pyplot as plt plt.subplot(311) plt.imshow(interp1.T) plt.colorbar() plt.subplot(312) plt.imshow(interp1_predicted.T) plt.colorbar() plt.subplot(313) plt.imshow((interp1_predicted - interp1).T) plt.colorbar() plt.show() # Control the values (with a precision of 3%) assert np.allclose(interp1_predicted, interp1, atol=3.e-2 * interp1.max())
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/')
n = np.where(z >= ramp_up + plateau + ramp_down, 0, n) # Add transverse guiding parabolic profile n = n * (1. + rel_delta_n_over_w2 * r**2) return (n) # The bunch bunch_zmin = z0 - 10.e-6 bunch_zmax = bunch_zmin + 4.e-6 bunch_rmax = 10.e-6 bunch_gamma = 400. bunch_n = 5.e23 # Convert parameters to boosted frame bunch_beta = np.sqrt(1. - 1. / bunch_gamma**2) bunch_zmin, bunch_zmax = \ boost.copropag_length( [ bunch_zmin, bunch_zmax ], beta_object=bunch_beta ) bunch_n, = boost.copropag_density([bunch_n], beta_object=bunch_beta) bunch_gamma, = boost.gamma([bunch_gamma]) # The moving window (moves with the group velocity in a plasma) v_window = c * (1 - 0.5 * n_e / 1.75e27) # Convert parameter to boosted frame v_window, = boost.velocity([v_window]) # The diagnostics diag_period = 50 # Period of the diagnostics in number of timesteps # Whether to write the fields in the lab frame Ntot_snapshot_lab = 20 dt_snapshot_lab = (zmax - zmin) / c track_bunch = False # Whether to tag and track the particles of the bunch
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()
# Create empty arrays for saving rms bunch sizes sig_zp = np.zeros(int(N_step / N_show) + 1) sig_xp = np.zeros(int(N_step / N_show) + 1) sig_yp = np.zeros(int(N_step / N_show) + 1) # Set initial bunch sizes sig_zp[0] = np.std(sim.ptcl[0].z) sig_xp[0] = np.std(sim.ptcl[0].x) sig_yp[0] = np.std(sim.ptcl[0].y) # Create array corresponding to the propagation distance of the bunch z_prop = np.arange(int(N_step / N_show) + 1) * ((zmax - zmin) / Nz) * N_show z_prop += -20.e-6 if l_boost: z_prop, = boost.copropag_length([z_prop], beta_object=boost.beta0) # Carry out the simulation for k in range(int(N_step / N_show)): sim.step(N_show) sig_zp[k + 1] = np.std(sim.ptcl[0].z) sig_xp[k + 1] = np.std(sim.ptcl[0].x) sig_yp[k + 1] = np.std(sim.ptcl[0].y) if show_fields: # Show the fields plt.figure(0) plt.clf() sim.fld.interp[0].show('Ez') plt.figure(1) plt.clf() sim.fld.interp[0].show('Er')
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/')