LEC DocumentationRelease 0.0.1

Ong Jian Fuh

Jul 21, 2020

INSTALLATION

1 Installation 31.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Dependencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 QuickStart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Usage 72.1 Input parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Output files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 Python . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.2 Gnuplot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.1 Single electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.2 Electron bunch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.3 Load particle from file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4.1 Landau-Liftshitz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4.2 Sokolov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4.3 Stochastic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4.4 Quantum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4.5 Emission cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Developers 173.1 Developers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Citation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Bibliography 19

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Welcome to the LEC guide. This is the documentation for LEC code, a particle code for laser-electron collision.It describes the dynamics of a relativistic electron bunch in the laser field based on Lorentz force equation.

This documentation is prepared for users with a minimum requirement of computational knowledge i.e. commandline in unix operating system. For the user who is fluent in Fortran90 and scripting knowledge, feel free to jumpto the respective sections.

Warning: This code is still in active development. Input and output may change from time to time.

INSTALLATION 1

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2 INSTALLATION

CHAPTER

ONE

INSTALLATION

1.1 Overview

This simulation was developed to simulate the laser-electron bunch collision and written in Fortran. The electronsare described by weighted computational particles and their motion are governed by Lorentz force equation. Thelaser energy is assumed to be constant, which is true unless the electron charge is in the order of nanoCoulomb.Therefore, Maxwell’s equations are not solved and the laser fields are treated as functions in space and time. Theparticle and laser have three-dimensional components, i.e., 𝑥, 𝑦, 𝑧, 𝑣𝑥,𝑦,𝑧, 𝐸,𝐵𝑥,𝑦,𝑧 . Additional features for laserand electron are available. The simulation setup is depicted as follow:

Note!

The electron bunch is treated as a two-dimensional sheath. This approximation is valid for near head-on collision.The longitudinal bunch dimension is not supported at the moment.

1.2 Features

• Radiation Reaction (RR)

– Landau-Liftshitz (LL)

– Sokolov (SKL)

– Stochastic (SCS)

• Classical and QED-assisted model for LL & SKL

• Radiation model

– Nonlinear Compton Scattering

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• Simulating a single electron or an electron bunch

• Calculating radiation spectrum and photon number distribution

• Calculating radiation angular distribution

• Pair production

– Bethe-Heitler

– nonlinear Breit-Wheeler (to do)

• Laser field beyond paraxial approximation up to fifth order

• Spatial and temporal Gaussian shape laser pulse (Linear & Circular polarisation)

• Load electron beam energy distribution from experimental data

1.3 Dependencies

The initial program format was FORTRAN77 (CONTINUE, GO TO etc.) with fortran90 features (modules,allocatable array etc.). Standard functions of Massage Passing Interface (MPI) and OpenMP are used. To installthe code, the following dependencies are needed in your PC as well as cluster. Please refer to the links below forthe installation of these dependencies.

• fortran compiler (e.g. gfortran)

– Intel Fortran compiler is not supported yet.

• Massage Passing Interface, MPI (e.g. OpenMPI)

• OpenMP

– OpenMP is shipped along with GCC 4.2 and above.

1.4 QuickStart

This code was written on unix based operating system such as macOS and Linux. Installation and execution areperformed by using command line. Download the code with git command:

git clone https://github.com/StevE-Ong/LEC.git

or use the Download Zip button. In the command line, change to the code directory and compile the code by:

./LEC.cmp

This script run the following command:

#!/bin/csh -xset COMPILER=mpif90set dir=$PWDset SRC0=$dir/src/LEC.fset SRC1=$dir/src/laser.fset LOADM=$dir/bin/LECset OBJDIR=$dir/obj

mkdir $dir/obj/mkdir $dir/bin/

##$COMPILER -O3 -fopenmp -fopt-info-all -mcmodel=medium -fconvert=big-endian -g

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-fbacktrace -fbounds-check -J$OBJDIR -o $LOADM $SRC0 $SRC1 >& cmp.lst#echo " ------------ End of Compile -------------- "#

The compiler option -fopenmp is required for programs with OpenMP functions. The option -fbounds-check is used to detect segmentation errors. This option may be excluded except for debug run. The option -O3specifies the third level of optimisation. The option -o specify the name of the executable file. Other complileroptions are listed here. If compilation is completed successfully, an executable file LEC is generated and locatedin /bin. The file cmp.lst is the compilation log file. Any code error for an unsuccessful compilation will bewritten here. Example of cmp.lst file:

/home/ong/LEC/src/LEC.f:648:9:

end1

Error: Expecting END SUBROUTINE statement at (1)

To run the simulation:

echo Data | mpirun -np 32 ./bin/LEC

where Data is a folder (can be a folder path, e.g. /examples/Data1) contains the input file input.dat. Thempirun command is used to run the executable file with MPI library. The option -n or -np specifying thenumber of MPI processes (i.e. 32 processes). The number of threads can be specified as

export OMP_NUM_THREADS=4

In this case the number of threads is 4.

To run multiple simulations at one execution, the output files with each input are prepared. In this examples theoutput files are located in /examples/Data1 and /examples/Data2. The output directory can be changedto your own/preferred directory followed by $i without space. For example:

#!/bin/csh

set i = 1

while($i<3)echo "Running simulation "$iecho examples/Data$i | mpirun -np 1 ./bin/LEC@ i++

end

Then run the simulation:

./LEC_multirun.jcf

The following output will be displayed in the command line. In this case, 1 MPI process is used with 64 threads.Radiation emission is calculated. The calculation will terminate with final runtime indicated. If there are morethan one simulation, the run will continue with similar output.

Running simulation 1Specify output directoryOutput directory:examples/Data1/

###L ########E #######C###L ########E ##########C###L ###E ###C ###C

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###L ########E ###C###L ########E ###C###L ###E ###C ###C#######L ########E ##########CLASER### ELECTRON# COLLISION

Welcome to Laser Electron Collision code (v-1.3.0)

*****************************************************The code is running on 1 processors

64 threads

*****************************************************Produce radiation

Iteration = 1000 ; Time step = 2.3852819683908048E-017Iteration = 2000 ; Time step = 4.7705639367816095E-017Iteration = 3000 ; Time step = 7.1558459051724140E-017...Iteration = 9998000 ; Time step = 2.3848049119971265E-013Iteration = 9999000 ; Time step = 2.3850434401939656E-013Iteration = 10000000 ; Time step = 2.3852819683908046E-013Calculating angular distribution...calculating radiationFinal runtime = 36 secondsRunning simulation 2Specify output directoryOutput directory:examples/Data2/..

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CHAPTER

TWO

USAGE

2.1 Input parameters

Note!

New PARAM may be added from time to time.

This section describes the details of the input parameters, output files, and examples. The code was written infortran90. The input parameters are contained in the file input.dat in the folder Data.

&PARAM1 jobno=001,ksmax=1000000,div=60.d0,div2=40.d0 ,&END&PARAM2 SL=1.0d22,Ev=10.d6,pw=0.8d-6,pp=5.6d-6,sp=1.44d-6 ,&END&PARAM3 alpha=0.0d-6,enum=1.0d9,bin=1.d6,shot=1,inc_ang = 0.d0 ,&END&PARAM4 xinit=1.1d0,rmass=1.d0,sigmax=0.01d0,sigmay=0.01d0,sigmaz=0.01d0 ,&END&PARAM5 iconR=2,QED=1,ipl=0,shape=0,OutRad=0,OutPairs=0 ,&END&PARAM6 loadpar=1,loadseed=33587,qedseed=0 ,&END

jobno [Integer] The job numbering.

ksmax [Integer] Maximum number of time step.

div,div2 [Float] The number of division or cells per Lamor radius.

SL [Float] Laser intensity [W cm−2].

Ev [Float] Electron energy [eV].

pw [Float] Laser wavelength [meter].

pp [Float] Laser pulse length [meter]. Pulse duration is 𝜏L = pp/𝑐. Pulse duration at FWHM is 𝜏L × 1.1744

sp [Float] Laser waist radius (at 1/𝑒2) [meter].

alpha [Float] Electron waist radius (at 1/𝑒2) [meter]. Set to 0.0d-6 for a single electron. If set to a value largerthan 0, electron energy distribution in 1D and 2D will be outputted.

enum [Float] Electron number in a bunch. Used for radiation calculation. For single electron, enum=1.d0

bin [Float] Radiation spectrum bin size [eV].

shot [Integer] Number of shot.

inc_ang [Float] Incident angle of the electron with respect to the x-axis [degree].

xinit [Float] Initial position of the electron from the collision point. The collision happens at 𝑡 = 0. The laser isone pulse length away from 𝑡 = 0.

rmass [Float] The charge to mass ratio of the colliding particle. Setting to rmass=206 represents muon.

sigma_{x,y,z} [Float] Electron momentum energy spreads in three directions. This input is ignored for a singleelectron

iconR [Integer] Specifying the particle pusher used.

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• iconR=0 Lorentz force

• iconR=1 Sokolov

• iconR=2 Reduced Landau-Liftshitz

• iconR=3 Stochastic

QED [Integer] Specifying whether to use QED for Sokolov and Reduced Landau-Liftshitz. QED=1 is mandatoryfor Stochastic process.

ipl [Integer] Specifying laser polarisation.

• ipl=0 linear polarisation (LP)

• ipl=1 circular polarisation (CP).

shape [Integer] Laser spatial and temporal profile. Laser profile can be modified in laser.f.

• shape=0 0th order Gaussian beam.

• shape=1 5th order paraxial approximation.

Note!

The temporal profile for 5th order paraxial approximation is not Gaussian. The temporal profile is 𝑔 = cosech((𝑡−𝑥)/𝜏L).

loadpar [Integer] Load particle energy distribution from external file. The default filename is dist_fn.dat.To change the filename, please edit the subroutine manual_load.

OutRad [Integer] Specifying whether to calculate radiation. When setting OutRad=1, emission spectrum, pho-ton number distribution, radiation angular distribution will be calculated. This part consumes most of the simula-tion time.

OutPairs [Integer] Specifying whether to calculate pair production. The code currently support the Bethe-Heitlerpair production. The cross section for Bremsstrahlung and pair production will be calculated if OutPairs=1.The Z component of nucleus for the specific converter is specify in module random_commom.

loadseed, [Integer] Specifying the user define seed for the loading of momentum distribution. If set to zero thenthe default seed in the code will be used.

qedseed, [Integer] Specifying the user define seed for Stochastic process (iconR=3). If set to zero then thedefault seed in the code will be used. To obtain different stochastic event of the same electron bunch, keeploadseed fixed and change qedseed.

2.2 Output files

Example of a single electron for Lorentz vs Sokolov. The input parameters are in examples/Data1(2)

&PARAM1 jobno=001,ksmax=10000000,div=128.d0,div2=40.d0 ,&END&PARAM2 SL=1.0d22,Ev=100.d6,pw=0.8d-6,pp=5.6d-6,sp=1.44d-6 ,&END&PARAM3 alpha=0.0d0,enum=1.0d0,bin=1.d6,shot=1,inc_ang = 0.d0 ,&END&PARAM4 xinit=5.d0,rmass=1.d0,sigmax=0.01d0,sigmay=0.01d0,sigmaz=0.01d0 ,&END&PARAM5 iconR=0,QED=0,ipl=0,shape=0,load=0,OutRad=1,OutPairs=0 ,&END

&PARAM1 jobno=002,ksmax=10000000,div=128.d0,div2=40.d0 ,&END&PARAM2 SL=1.0d22,Ev=100.d6,pw=0.8d-6,pp=5.6d-6,sp=1.44d-6 ,&END&PARAM3 alpha=0.0d0,enum=1.0d0,bin=1.d6,shot=1,inc_ang = 0.d0 ,&END&PARAM4 xinit=5.d0,rmass=1.d0,sigmax=0.01d0,sigmay=0.01d0,sigmaz=0.01d0 ,&END&PARAM5 iconR=1,QED=0,ipl=0,shape=0,load=0,OutRad=1,OutPairs=0 ,&END

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Note!

New PARAM may be added from time to time. Please refers to the input list to see the lastest PARAM.

The outputs are written in ASCII format. The file output001.dat records the detail parameters of the simula-tion. For example:

Parameters for pulse laser

Laser polarization: linear

0th order Gaussian beam

Laser Intensity [W/cm2] 1.0000000000000000E+022Peak electric field [V/m] 274000000000000.00Peak magnetic field [Gauss] 9280000000.0000000Larmor radius for light speed [m] 1.8318965517241380E-009laser wavelength [m] 7.9999999999999996E-007pulse length [m] 5.5999999999999997E-006pulse duration [s] 1.8666666666666665E-014pulse duration (FWHM) [s] 2.1978133333333330E-014waist radius (1/e2) [m] 1.4400000000000000E-006

parameters for electron beam...

The file orbt1q001.dat records the trajectories, energy etc. of the particle. For a single electron, there are 7files recoding the same output. For example:

0 1 2 3 4 5 6time [s] x [m] y [m] 𝑝𝑥 [normalized] 𝑝𝑦 [normalized] kinetic energy [eV] 𝜒𝑒 [dimensionless]

-0.466547E-13 0.279928E-04 -0.203395E-54 -0.196692E+03 -0.168832E-43→˓ 0.100511E+09 0.360270E-45-0.466428E-13 0.279857E-04 -0.166512E-53 -0.196692E+03 -0.696642E-43→˓ 0.100511E+09 0.753717E-45-0.466309E-13 0.279785E-04 -0.574687E-53 -0.196692E+03 -0.161358E-42→˓ 0.100511E+09 0.117740E-44-0.466190E-13 0.279714E-04 -0.139148E-52 -0.196692E+03 -0.294680E-42→˓ 0.100511E+09 0.162775E-44-0.466070E-13 0.279642E-04 -0.277271E-52 -0.196692E+03 -0.471967E-42→˓ 0.100511E+09 0.210042E-44-0.465951E-13 0.279571E-04 -0.488190E-52 -0.196692E+03 -0.695102E-42→˓ 0.100511E+09 0.259021E-44...

Note!

The numbers 0, 1, 2,. . . indicate the columns to extract the data by using usecols=[0,1,2,. . . ] in Python. Forgnuplot, the columns number becomes ($1), ($2), ($3),. . .

The file phtne001.dat records the radiation output. For example:

0 1 2energy [eV] photon number photon number × energy

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8333.3333333333321 238.86944345907790 6.263351361621747825000.000000000000 182.50447244093024 7.550733693220039741666.666666666664 104.14244180601469 9.138052900480476058333.333333333328 89.422617071344263 9.857999556749812075000.000000000000 70.619159337234422 10.69747696935665591666.666666666657 63.363841302196569 11.199199843048401...

The file phtnTe001.dat records the radiation angular distribution. For example:

0 1 2𝜃𝑧 [rad] 𝜃𝑦 [rad] radiated energy [a.u]

-0.3139E+01 -0.3139E+01 0.0000E+00-0.3132E+01 -0.3139E+01 0.0000E+00-0.3126E+01 -0.3139E+01 0.0000E+00-0.3120E+01 -0.3139E+01 0.0000E+00

For an electron bunch, there are more than 7 outputs, depending on the number of MPI processes. Eachoutput record a sample electron information. On the other hand, file such as AveEne(jobno).dat,dist_fn(kstep)(jobno).dat, dist_fn2d(kstep)(jobno).dat will be output.

The file AveENE records:

0 1 2 3time [s] average kinetic energy [eV] average + 𝜎+ [eV] average - 𝜎− [eV]

The meaning of 𝜎± is slightly different from standard deviation. It is defined as:

𝜎± =

∑︀|𝛾𝑖 − ⟨𝛾⟩|±𝑁±

where 𝜎+ is the energy deviation above the average while 𝜎− is the energy deviation below the average. 𝑁± isthe number of particles above and below the average energy. In most circumstances, 𝜎+ = 𝜎− and we can saythat the particle energy has a normal distribution. In some cases, a deviation from normal distribution after theinteraction may be observed.

The file dist_fn records:

0 1energy [eV] electron number [a.u]

The file dist_fn2d records:

0 1 2𝑝𝑦 [normalized] 𝑝𝑧 [normalized] electron number [a.u]

2.2.1 Python

In this examples, the visualisation is performed by using Python in Jupyter notebook. The python codes can befound in /examples/**.ipynb. The extension .ipynb stand for Jupiter notebook. The data can be read asfollows:

#Time evolution of electron energyT1,x1,y1,px1,py1,E1,Xi1 = np.loadtxt(rf"{run_dir}/examples/Data1/orbt1q"

+str(file1).zfill(3)+".dat",unpack=True,

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usecols=[0,1,2,3,4,5,6],dtype=np.float)T2,x2,y2,px2,py2,E2,Xi2 = np.loadtxt(rf"{run_dir}/examples/Data2/orbt1q"

+str(file2).zfill(3)+".dat",unpack=True,usecols=[0,1,2,3,4,5,6],dtype=np.float)

In the Jupyter notebook, there is a python function import figformat. This function output/display figureswith selected parameters. The figure width, fig_width is set to 3.4 inches, represents a single column width of adouble column journal.

import matplotlib as mplimport figformatfig_width,fig_height,params=figformat.figure_format(fig_width=3.4,fig_height=2)mpl.rcParams.update(params)

The figure width can be override to any number by writing fig.set_size_inches(fig_width*2,fig_width/1.618) at each plot. The number 1.618 is the Golden ratio. Multiplying or dividing thefig_width by the Golden ratio for figure height ensure the nice appearance of a figure. Other parameters suchas font size, plot line width, ticks width and etc. can be changed in the file figformat.py.

2.2.2 Gnuplot

On the other hand, a quick visualisation can be performed by using gnuplot. For example:

> plot “***.dat” using ($1):($4) with lines> replot “***.dat” using ($1):($4) with lines

2.3 Examples

2.3.1 Single electron

In this example, we plot several outputs of a single electron. Details of the plotting code can be referred to theJupyter notebook. It can be viewed in GitHub. We showed the output for Lorentz (without RR) and Sokolov (withRR) in classical regime.

The electron trajectory

The time evolution of electron energy

The radiation spectrum

The photon number distribution

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2.3.2 Electron bunch

In this examples, we show the results of 109 electrons colliding with the laser with intensity 1022 Wcm−2. Theinput is:

&PARAM1 jobno=003,ksmax=1000000,div=60.d0,div2=40.d0 ,&END&PARAM2 SL=1.d22,Ev=600.d6,pw=0.82d-6,pp=3.3d-6,sp=5.5d-6 ,&END&PARAM3 alpha=1.d-6,enum=1.0d9,bin=1.d6,shot=1,inc_ang = 0.d0 ,&END&PARAM4 xinit=2.d0,rmass=1.d0,sigmax=0.1d0,sigmay=0.01d0,sigmaz=0.01d0 ,&END&PARAM5 iconR=1,QED=0,ipl=0,shape=0,OutRad=1,OutPairs=0 ,&END&PARAM6 loadpar=0,loadseed=33587,qedseed=0 ,&END

The longitudinal momentum spread is 10% of its initial kinetic energy, i.e. sigmax=0.1d0. Other componentsare set to a very small value. The simulations were run for Sokolov (classical, iconR=1, QED=0), Sokolov(QED-assisted, iconR=1, QED=1), and Stochastic (iconR=3, QED=1). For Stochastic, QED=1 is manda-tory.

2.3.3 Load particle from file

The particle energy distribution can be loaded from the output of PIC simulation. The distribution function filename is load_particle.dat. The data in the first column is the energy in the unit of electron volt [eV]. Thesecond column is the particle distribution. The value of the distribution can be scaled accordingly.

2.4 Models

Todo: To do

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Details numerical implementation can be found in Ref. [OTMT16].

2.4.1 Landau-Liftshitz

𝑑𝑣𝜇

𝑑𝜏=

𝑒

𝑚𝑐𝐹𝜇𝜈𝑣𝜈 + 𝜏0

(︂𝑒

𝑚𝑐�̇�𝜇𝜈𝑣𝜈 +

𝑒2

𝑚2𝑐2𝐹𝜇𝜈𝐹𝛼𝜈𝑣

𝛼 𝑒2

𝑚2𝑐2(𝐹𝛼𝜈𝑣𝜈)(𝐹𝛼𝜆𝑣

𝜆)𝑣𝜇)︂

2.4.2 Sokolov

𝑑𝑝𝜇

𝑑𝜏=

𝑒

𝑚𝑐𝐹𝜇𝜈𝑣𝜈 − 𝐼𝑄𝐸𝐷

𝑚𝑐2𝑝𝜇 + 𝜏0

𝑒2

(𝑚𝑐)2𝐼𝑄𝐸𝐷

𝐼𝐸𝐹𝜇𝜈𝐹𝜈𝛼𝑝

𝛼

2.4.3 Stochastic

2.4.4 Quantum

2.4.5 Emission cross-section

𝑑𝑊𝑒𝑚 =𝛼𝑚𝑐2√3𝜋~𝛾

[︂(︂1 − 𝜉 +

1

1 − 𝜉

)︂𝐾2/3(𝛿) −

∫︁ ∞

𝛿

𝐾1/3(𝑠)𝑑𝑠

]︂𝑑𝜉

𝜉 =~𝜔

𝛾𝑚𝑐2, 𝛿 =

2𝜉

3(1 − 𝜉)𝜒

and 𝐾𝜈(𝑥) is modified Bessel function. At classical limit 𝜒 << 1

𝑑𝑃 =

ℰ𝑑𝑊𝑒𝑚

→𝑒2𝜔𝑐√

3𝜋𝑐

1

𝛾2

𝜔

𝜔 𝑐[2𝐾2/3(𝛿) −

∫︁ ∞

𝛿

𝐾1/3(𝑠)𝑑𝑠]𝑑𝜔

reduced to classical synchrotron radiation where 𝜔𝑐 is the critical frequency and 𝛿 −→ 2𝜉/3𝜒.

The function 𝑞(𝜒𝑒) for 𝜒𝑒 ≪ 1 (blue)

𝑞(𝜒𝑒 ≪ 1) ≈ 1 − 55

16

√3𝜒 + 48𝜒2

The function 𝑞(𝜒𝑒) for 𝜒𝑒 ≫ 1 (green)

𝑞(𝜒𝑒 ≫ 1) ≈ 48

243Γ(

2

3)𝜒−4/3

[︂1 − 81

16Γ(2/3)(3𝜒)−2/3

]︂

2.4. Models 15

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16 Chapter 2. Usage

CHAPTER

THREE

DEVELOPERS

3.1 Developers

This code was originally written by T. Moritaka and extended by J. F. Ong

The first results of this code were published in Ref [OTMT16]. The code was then extended to particle-in-cell(PIC) simulation [OMT18][OMT19]. The code in Ref. [MBL+13] is not for laser-beam interaction but a routineto convert gamma photon from particle to mesh is partially relevant to the present code.

3.2 Citation

To cite this work, use the following BibTeX code:

@Article{LEC2016,author = {J. F. Ong and W. R. Teo and Toseo Moritaka and H. Takabe},title = {Radiation reaction in the interaction of ultraintense

laser with matter and gamma ray source},journal = {Physics of Plasmas},volume = {23},number = {5},pages = {053117},year = {2016},doi = {10.1063/1.4952626},url = { http://dx.doi.org/10.1063/1.4952626}}

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18 Chapter 3. Developers

BIBLIOGRAPHY

[MBL+13] Toseo Moritaka, Luca Baiotti, An Lin, Li Weiwu, Youichi Sakawa, Yasuhiro Kuramitsu, TaichiMorita, and Hideaki Takabe. Plasma particle-in-cell simulations with QED reactions for pair produc-tion experiments using a high-z solid target. Journal of Physics: Conference Series, 454:012016, aug2013. URL: https://doi.org/10.1088%2F1742-6596%2F454%2F1%2F012016, doi:10.1088/1742-6596/454/1/012016.

[OMT18] J. F. Ong, T. Moritaka, and H. Takabe. The suppression of radiation reaction and laser field depletionin laser-electron beam interaction. Physics of Plasmas, 25(3):033113, 2018. URL: https://doi.org/10.1063/1.5012937, doi:10.1063/1.5012937.

[OMT19] J. F. Ong, T. Moritaka, and H. Takabe. Optimizing the energies conversion in laser-electronbeam collision. Physics of Plasmas, 26(3):033102, 2019. URL: https://doi.org/10.1063/1.5054856,doi:10.1063/1.5054856.

[OTMT16] J. F. Ong, W. R. Teo, Toseo Moritaka, and H. Takabe. Radiation reaction in the interaction of ul-traintense laser with matter and gamma ray source. Physics of Plasmas, 23(5):053117, 2016. URL:http://dx.doi.org/10.1063/1.4952626, doi:10.1063/1.4952626.

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