The spectrally resolved far field radiation of charged macro particles.

Our simulation computes the Lienard Wiechert potentials to calculate the emitted electromagnetic spectra for different observation directions using the far field approximation.

$\frac{\operatorname{d}^2I}{\operatorname{d}{\Omega}\operatorname{d}\omega}\left(\omega,\vec{n}\right)=\left|\sum\limits_{k=1}^{N}\int\limits_{-\infty}^{+\infty}\frac{\vec{n}\times\left[\left(\vec{n}-\vec{\beta}_k(t)\right)\times\dot{\vec{\beta}}_k(t)\right]}{\left(1-\vec{\beta}_k(t)\cdot\vec{n}\right)^2}\cdot\operatorname{e}^{\operatorname{i}\omega\left(t-\vec{n}\cdot\vec{r}_k(t)/c\right)}\operatorname{d}t\right|^2$

Variable

Meaning

$$\vec r_k(t)$$

The position of particle k at time t.

$$\vec \beta_k(t)$$

The normalized speed of particle k at time t. (Speed divided by the speed of light)

$$\dot{\vec{\beta}}_k(t)$$

The normalized acceleration of particle k at time t. (Time derivative of the normalized speed.)

$$t$$

Time

$$\vec n$$

Unit vector pointing in the direction where the far field radiation is observed.

$$\omega$$

The circular frequency of the radiation that is observed.

$$N$$

Number of all (macro) particles that are used for computing the radiation.

$$k$$

Running index of the particles.

Currently this allows to predict the emitted radiation from plasma if it can be described by classical means. Not considered are emissions from ionization, Compton scattering or any bremsstrahlung that originate from scattering on scales smaller than the PIC cell size.

## External Dependencies¶

The plugin is available as soon as the libSplash and HDF5 libraries are compiled in.

## .param files¶

In order to setup the radiation analyzer plugin, both the radiation.param and the radiationObserver.param have to be configured and the radiating particles need to have the attribute momentumPrev1 which can be added in speciesDefinition.param.

In radiation.param, the number of frequencies N_omega and observation directions N_theta is defined.

### Frequency range¶

The frequency range is set up by choosing a specific namespace that defines the frequency setup

/* choose linear frequency range */


Currently you can choose from the following setups for the frequency range:

namespace

Description

linear_frequencies

linear frequency range from SI::omega_min to SI::omega_max with N_omega steps

log_frequencies

logarithmic frequency range from SI::omega_min to SI::omega_max with N_omega steps

frequencies_from_list

N_omega frequencies taken from a text file with location listLocation[]

All three options require variable definitions in the according namespaces as described below:

For the linear frequency scale all definitions need to be in the picongpu::plugins::radiation::linear_frequencies namespace. The number of total sample frequencies N_omega need to be defined as constexpr unsigned int. In the sub-namespace SI, a minimal frequency omega_min and a maximum frequency omega_max need to be defined as constexpr float_64.

For the logarithmic frequency scale all definitions need to be in the picongpu::plugins::radiation::log_frequencies namespace. Equivalently to the linear case, three variables need to be defined: The number of total sample frequencies N_omega need to be defined as constexpr unsigned int. In the sub-namespace SI, a minimal frequency omega_min and a maximum frequency omega_max need to be defined as constexpr float_64.

For the file-based frequency definition, all definitions need to be in the picongpu::plugins::radiation::frequencies_from_list namespace. The number of total frequencies N_omega need to be defined as constexpr unsigned int and the path to the file containing the frequency values in units of $$\mathrm{[s^{-1}]}$$ needs to be given as constexpr const char * listLocation = "/path/to/frequency_list";. The frequency values in the file can be separated by newlines, spaces, tabs, or any other whitespace. The numbers should be given in such a way, that c++ standard std::ifstream can interpret the number e.g., as 2.5344e+16.

Note

Currently, the variable listLocation is required to be defined in the picongpu::plugins::radiation::frequencies_from_list namespace, even if frequencies_from_list is not used. The string does not need to point to an existing file, as long as the file-based frequency definition is not used.

### Observation directions¶

The number of observation directions N_theta is defined in radiation.param, but the distribution of observation directions is given in radiationObserver.param) There, the function observation_direction defines the observation directions.

This function returns the x,y and z component of a unit vector pointing in the observation direction.

DINLINE vector_64
observation_direction( int const observation_id_extern )
{
/* use the scalar index const int observation_id_extern to compute an
* observation direction (x,y,y) */
return vector_64( x , y , z );
}


Note

The radiationObserver.param set up will be subject to further changes. These might be namespaces that describe several preconfigured layouts or a functor if C++ 11 is included in the nvcc.

### Nyquist limit¶

A major limitation of discrete Fourier transform is the limited frequency resolution due to the discrete time steps of the temporal signal. (see Nyquist-Shannon sampling theorem) Due to the consideration of relativistic delays, the sampling of the emitted radiation is not equidistantly sampled. The plugin has the option to ignore any frequency contributions that lies above the frequency resolution given by the Nyquist-Shannon sampling theorem. Because performing this check costs computation time, it can be switched off. This is done via a precompiler pragma:

// Nyquist low pass allows only amplitudes for frequencies below Nyquist frequency
// 1 = on (slower and more memory, no Fourier reflections)
// 0 = off (faster but with Fourier reflections)
#define __NYQUISTCHECK__ 0


Additionally, the maximally resolvable frequency compared to the Nyquist frequency can be set.

namespace radiationNyquist
{
/* only use frequencies below 1/2*Omega_Nyquist */
const float NyquistFactor = 0.5;
}


This allows to make a save margin to the hard limit of the Nyquist frequency. By using NyquistFactor = 0.5 for periodic boundary conditions, particles that jump from one border to another and back can still be considered.

### Form factor¶

The form factor is a method, which considers the shape of the macro particles when computing the radiation.

One can select between different macro particle shapes. Currently eight shapes are implemented. A shape can be selected by choosing one of the available namespaces:

/* choosing the 3D CIC-like macro particle shape */


Namespace

Description

radFormFactor_CIC_3D

3D Cloud-In-Cell shape

radFormFactor_TSC_3D

3D Triangular shaped density cloud

radFormFactor_PCS_3D

3D Quadratic spline density shape (Piecewise Cubic Spline assignment function)

radFormFactor_CIC_1Dy

Cloud-In-Cell shape in y-direction, dot like in the other directions

radFormFactor_Gauss_spherical

symmetric Gauss charge distribution

radFormFactor_Gauss_cell

Gauss charge distribution according to cell size

radFormFactor_incoherent

forces a completely incoherent emission by scaling the macro particle charge with the square root of the weighting

radFormFactor_coherent

forces a completely coherent emission by scaling the macro particle charge with the weighting

### Reducing the particle sample¶

In order to save computation time, only a random subset of all macro particles can be used to compute the emitted radiation. In order to do that, the radiating particle species needs the attribute radiationMask (which is initialized as false) which further needs to be manipulated, to set to true for specific (random) particles.

Note

The reduction of the total intensity is not considered in the output. The intensity will be (in the incoherent case) will be smaller by the fraction of marked to all particles.

Note

Note

In future updates, the radiation will only be computed using an extra particle species. Therefore, this setup will be subject to further changes.

### Gamma filter¶

In order to consider the radiation only of particles with a gamma higher than a specific threshold, the radiating particle species needs the attribute radiationMask (which is initialized as false). Using a filter functor as:

using RadiationParticleFilter = picongpu::particles::manipulators::FreeImpl<
GammaFilterFunctor
>;


(see Bunch or Kelvin Helmholtz example for details) sets the flag to true is a particle fulfills the gamma condition.

Note

More sophisticated filters might come in the near future. Therefore, this part of the code might be subject to changes.

### Window function filter¶

A window function can be added to the simulation area to reduce ringing artifacts due to sharp transition from radiating regions to non-radiating regions at the boundaries of the simulation box. This should be applied to simulation setups where the entire volume simulated is radiating (e.g. Kelvin-Helmholtz Instability).

In radiation.param the precompiler variable PIC_RADWINDOWFUNCTION defines if the window function filter should be used or not.

// add a window function weighting to the radiation in order
// to avoid ringing effects from sharp boundaries
// 1 = on (slower but with noise/ringing reduction)
// 0 = off (faster but might contain ringing)


If set to 1, the window function filter is used.

There are several different window function available:

/* Choose different window function in order to get better ringing reduction
*/



By setting radWindowFunction a specific window function is selected.

## .cfg file¶

For a specific (charged) species <species> e.g. e, the radiation can be computed by the following commands.

Command line option

Description

--<species>_radiation.period

Gives the number of time steps between which the radiation should be calculated. Default is 0, which means that the radiation in never calculated and therefor off. Using 1 calculates the radiation constantly. Any value >=2 is currently producing nonsense.

--<species>_radiation.dump

Period, after which the calculated radiation data should be dumped to the file system. Default is 0, therefor never. In order to store the radiation data, a value >=1 should be used.

--<species>_radiation.lastRadiation

If set, the radiation spectra summed between the last and the current dump-time-step are stored. Used for a better evaluation of the temporal evolution of the emitted radiation.

--<species>_radiation.folderLastRad

Name of the folder, in which the summed spectra for the simulation time between the last dump and the current dump are stored. Default is lastRad.

--<species>_radiation.totalRadiation

If set the spectra summed from simulation start till current time step are stored.

--<species>_radiation.folderTotalRad

Folder name in which the total radiation spectra, integrated from the beginning of the simulation, are stored. Default totalRad.

--<species>_radiation.start

Time step, at which PIConGPU starts calculating the radiation. Default is 2 in order to get enough history of the particles.

--<species>_radiation.end

Time step, at which the radiation calculation should end. Default: 0 (stops at end of simulation).

--<species>_radiation.radPerGPU

If set, each GPU additionally stores its own spectra without summing over the entire simulation area. This allows for a localization of specific spectral features.

--<species>_radiation.folderRadPerGPU

Name of the folder, where the GPU specific spectra are stored. Default: radPerGPU

--<species>_radiation.compression

If set, the hdf5 output is compressed.

## Memory Complexity¶

### Accelerator¶

each energy bin times each coordinate bin allocates one counter (float_X) permanently and on each accelerator.

### Host¶

as on accelerator.

## Output¶

Depending on the command line options used, there are different output files.

Command line flag

Output description

--<species>_radiation.totalRadiation

Contains ASCII files that have the total spectral intensity until the timestep specified by the filename. Each row gives data for one observation direction (same order as specified in the observer.py). The values for each frequency are separated by tabs and have the same order as specified in radiation.param. The spectral intensity is stored in the units $$\mathrm{[Js]}$$.

--<species>_radiation.lastRadiation

has the same format as the output of totalRadiation. The spectral intensity is only summed over the last radiation dump period.

--<species>_radiation.radPerGPU

Same output as totalRadiation but only summed over each GPU. Because each GPU specifies a spatial region, the origin of radiation signatures can be distinguished.

In the folder radiationHDF5, hdf5 files for each radiation dump and species are stored. These are complex amplitudes in units used by PIConGPU. These are for restart purposes and for more complex data analysis.

### Text-based output¶

The text-based output of lastRadiation and totalRadiation contains the intensity values in SI-units $$\mathrm{[Js]}$$. Intensity values for different frequencies are separated by spaces, while newlines separate values for different observation directions.

In order to read and plot the text-based radiation data, a python script as follows could be used:

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.colors import LogNorm

# frequency definition:
# as defined in the 'radiation.param' file:
N_omega = 1024
omega_min = 0.0 # [1/s]
omega_max = 5.8869e17 # [1/s]
omega = np.linspace(omega_min, omega_max, N_omega)

# observation angle definition:
# as defined in the 'radiation.param' file:
N_observer = 128
# as defined in the 'radiationObserver.param' file:
# this example assumes one used the default Bunch example
# there, the theta values are normalized to the Lorentz factor
theta = np.linspace(theta_min, theta_max, N_observer)

plt.figure()

cb = plt.colorbar()
cb.set_label(r"$\frac{\mathrm{d}^2 I}{\mathrm{d} \omega \mathrm{d} \Omega} \, \mathrm{[Js]}$", fontsize=18)
for i in cb.ax.get_yticklabels():
i.set_fontsize(14)

# configure x-axis
plt.xlabel(r"$\omega \, \mathrm{[1/s]}$", fontsize=18)
plt.xticks(fontsize=14)

# configure y-axis
plt.ylabel(r"$\theta / \gamma$", fontsize=18)
plt.yticks(fontsize=14)

# make plot look nice
plt.tight_layout()
plt.show()


### HDF5 output¶

The hdf5 based data contains the following data structure in /data/{iteration}/DetectorMesh/ according to the openPMD standard:

Amplitude (Group):

Dataset

Description

Dimensions

x_Re

real part, x-component of the complex amplitude

(N_observer, N_omega, 1)

x_Im

imaginary part, x-component of the complex amplitude

(N_observer, N_omega, 1)

y_Re

real part, y-component of the complex amplitude

(N_observer, N_omega, 1)

y_Im

imaginary part, y-component of the complex amplitude

(N_observer, N_omega, 1)

z_Re

real part, z-component of the complex amplitude

(N_observer, N_omega, 1)

z_Im

imaginary part, z-component of the complex amplitude

(N_observer, N_omega, 1)

DetectorDirection (Group):

Dataset

Description

Dimensions

x

x-component of the observation direction $$\vec n$$

(N_observer, 1, 1)

y

y-component of the observation direction $$\vec n$$

(N_observer, 1, 1)

z

z-component of the observation direction $$\vec n$$

(N_observer, 1, 1)

DetectorFrequency (Group):

Dataset

Description

Dimensions

omega

frequency $$\omega$$ of virtual detector bin

(1, N_omega, 1)

Please be aware that all datasets in the hdf5 output are given in the PIConGPU-intrinsic unit system. In order to convert, for example, the frequencies $$\omega$$ to SI-units one has to multiply with the dataset-attribute unitSI.

import h5py
omega_handler = f['/data/2800/DetectorMesh/DetectorFrequency/omega']
omega = omega_handler[0, :, 0] * omega_handler.attrs['unitSI']
f.close()


In order to extract the radiation data from the HDF5 datasets, PIConGPU provides a python module to read the data and obtain the result in SI-units. An example python script is given below:

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.colors import LogNorm

# get frequencies

# get all observation vectors and convert to angle

gamma = 5.0
theta_norm = np.arctan(vec_n[:, 0]/vec_n[:, 1]) * gamma

# get spectrum over observation angle

plt.figure()
plt.pcolormesh(omega, theta_norm, spectrum, norm=LogNorm())

cb = plt.colorbar()
cb.set_label(r"$\frac{\mathrm{d}^2 I}{\mathrm{d} \omega \mathrm{d} \Omega} \, \mathrm{[Js]}$", fontsize=18)
for i in cb.ax.get_yticklabels():
i.set_fontsize(14)

# configure x-axis
plt.xlabel(r"$\omega \, \mathrm{[1/s]}$", fontsize=18)
plt.xticks(fontsize=14)

# configure y-axis
plt.ylabel(r"$\theta / \gamma$", fontsize=18)
plt.yticks(fontsize=14)

# make plot look nice
plt.tight_layout()
plt.show()


There are various methods besides get_Spectra() that are provided by the python module. If a method exists for _x (or _X) it also exists for _y and _z (_Y and _Z) accordingly.

Method

Description

.get_omega()

get frequency $$\omega$$ of virtual detector bin in units of $$\mathrm{[1/s]}$$

.get_vector_n()

get observation direction $$\vec{n}$$

.get_Spectra()

get spectrum $$\mathrm{d}^2 I / \mathrm{d} \omega \mathrm{d} \Omega$$ in units of $$\mathrm{[Js]}$$

.get_Polarization_X()

get spectrum but only for polarization in x-direction

.get_Amplitude_x()

get x-component of complex amplitude (unit: $$\mathrm{[\sqrt{Js}]}$$)

.get_timestep()

the iteration (timestep) at which the data was produced (unit: PIC-cycles)

Note

Modules for visualizing radiation data and a widget interface to explore the data interactively will be developed in the future.

## Analyzing tools¶

In picongp/src/tools/bin, there are tools to analyze the radiation data after the simulation.

Tool

Description

plotRadiation

Reads ASCII radiation data and plots spectra over angles as color plots. This is a python script that has its own help. Run plotRadiation --help for more information.

radiationSyntheticDetector

Reads ASCII radiation data and statistically analysis the spectra for a user specified region of observation angles and frequencies. This is a python script that has its own help. Run radiationSyntheticDetector --help for more information.

smooth.py

Python module needed by plotRadiation.

## Known Issues¶

The plugin supports multiple radiation species but spectra (frequencies and observation directions) are the same for all species.