7 

 This work is licensed under a Creative Commons Attribution 4.0 International License. 

 

  

 

 

 Simulation and Analysis the Effect of the Lorentz Force in a Free Electron 

Laser 

 

 

 

                         

                         

  

                                                      

 

 

Abstarct   

    Due to the scientific and technical development in the free electron laser devices and the 

accompanying industrial and technological progress in various fields of civil and military life, it 

became necessary to expand the understanding of the mechanism of interaction of electrons (as 

an effective medium) with the magnetic field that they pass through to form coherent photons. 

 

    In this paper, the Lorentz force effect is simulated and analysed. The results showed that the 

Lorentz force originates from the magnetic field, making the electron move through it oscillate. 

This sinusoidal motion of the electron causes it to emit two photons for every electron 

wavelength. It has been concluded that the electron velocity directly affects the Lorentz power 

and the wavelength and power of the output laser. 

 

Keywords free electron laser, Lorentz force, wavelength, power. 
 

1. Introduction  
    At the beginning of the seventies the last century, John Madey [1] invented the free electron 

laser FEL, a new type of laser, that differs in its mechanism of action from the common lasers.  

A free electron beam is passed through an opposite periodic magnetic field, which causes the 

electron to oscillate by the force of Lorentz, to produce coherent photons that make up the laser 

beam. 

Through a wide review of the literature dealing with the subject of the free electron laser, it was 

noted that the effect of the Lorentz force did not receive enough attention despite its great 

importance in making the electron's oscillatory motion and thus generating the coherent  

Ibn Al Haitham Journal for Pure and Applied Sciences 

Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index 

 

Doi: 10.30526/35.2.2775 

Thair Abdulkareem Khalil Al-Aish 

thair.ak.i@ihcoedu.uobaghdad.edu.iq 

Department of Physics, College of 

Education for Pure Sciences Ibn Al-

Haitham, University of Baghdad, 

Baghdad, Iraq. 

 

Article history: Received,16, January, 2022, Accepted,20, February, 2022, Published in April 2022. 

 

 

 

Hanady Amjed Kamil 

hanadyamjedkamil@gmail.com 

Directorate of Education of First 

Karkh, Ministry of Education, 

Baghdad, Iraq. 

 

https://creativecommons.org/licenses/by/4.0/
mailto:thair.ak.i@ihcoedu.uobaghdad.edu.iq
mailto:hanadyamjedkamil@gmail.com


Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 
 

8 

 

photons. Therefore, the focus of this work was on analyzing the Lorentz force in terms of its 

origin and its effect on the path of electron movement through the undulator [2-6].  

In this work, an executable program ELFFEL was constructed using Matlab 2019 software it as 

shown in Figure (1). contained several parameters to simulate the change in the movement 

electrons from accelerated linear motion to sinusoidal motion of the synchrotron beam 

formation. All factors affecting Lorentz force formation were studied. This was done by dividing 

the electron wave movement into regions that differed according to the direction of the magnetic 

field. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 1: The implementation of executable program ELFFEL to Simulation and analysis the effect of the Lorentz 

force in a free electron laser. 

 

2. Theory of the Free Electron Laser FEL 
    A free electron laser has an effective medium represented by a beam of electrons, which 

distinguishes it from other conventional lasers. That led to its possession of important 

advantages, including the ability to tune the wavelength of the external laser to cover a wide 

range of the electromagnetic spectrum, as well as the high power of the laser beam. The free 

electron laser consists of four main components, as shown in Figure (2).[7-10] 

* The Electrons gun. 

* The Linear accelerator for electrons. 

* The undulator is a set of magnets arranged oppositely. 

* The resonator is two mirrors, one of which is fully reflective and the other partially reflective. 

The electrons are launched from the gun to be accelerated through the linear accelerator. When 

entering the undulator, the magnetic field of the magnet will be affected by the force of Lorentz, 

which causes it to oscillate in a sine wave to emit the coherent photons and form the output laser 

beam, which will be explained later in detail. [10-13] 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 
 

9 

 

 
Figure 2. Components of free electron laser FEL 

After performing several calculations, the equations for the wavelength 𝜆 and power P of the 
output laser beam are derived as shown below:[2,6,7,8] 

𝜆 =  4.095 ×  10−14 × (
𝜆𝑢

𝐸𝑒
2

) (1 + (4354.77 × 𝜆𝑢
2

 𝐵2))                     (1) 

𝑃 =   1.67 × 1023  𝐸𝑒
2   𝑒𝑥𝑝

(2𝐿𝑟 +0.021(
𝐿𝑢
𝜆𝑢

))
                                                (2) 

Where:  𝜆𝑢  is the wavelength of the electron, E is the energy of the electrons beam; B is the 
magnetic field, 

𝐿𝑢 is the length of the undulator and 𝐿𝑟 is the length of the resonator. 
 

3. Results and discussion of Simulation  
    The electron is launched from the electronic gun to be accelerated by a linear accelerator to 

reach values 𝑣 close to the speed of light, then it enters the undulator towards the index finger 

according to the rule of the left hand as shown in Figure (3-a). 

 
Figure 3. The rule of the left hand 

    Figure (4) represents an undulator part divided into four regions to analyze and simulate the 

movement of an electron through two rows of magnets, arranged periodically and oppositely. 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 
 

10 

 

 
Figure 4. An electron oscillates through the undulator. 

    In region (1) of the undulator, the direction of the magnetic field 𝐵 is towards the thumb 

according to the rule of the left hand as shown in Figure (3-a), so the electron will be deflected 

as a result of being affected by the Lorentz force 𝐹. 

    Lorentz force 𝐹 acts on an electric charge 𝑞 moving in a magnetic field, which is discovered 

by Dutch scientist Hendrik Lorentz. In a magnetic field, the Lorentz force is the most significant 

when the direction of electron motion is perpendicular to the magnetic field lines. If the electron 

moves in a direction parallel to the direction of the magnetic field lines, the Lorentz force does 

not arise. The Lorentz force is always perpendicular to the direction of the electron's motion 𝑣 

and the magnetic field lines𝐵. The Lorentz force value is given by the following equation:[14-

20] 

𝐹 = 𝑞(𝑣 × 𝐵) = 𝑒𝑣𝐵 sin ∅                                    (3)  

 

    Where ∅ is the angle between the 𝑣 and 𝐵. The Lorentz force causes the electron to move in a 

sine wave due to the presence of the term (sin ∅ ). when the angle is 900 at point A, the Lorentz 

magnetic force will be at its greatest value, causing a large deviation in the electron's motion at 

the beginning of its entry into the undulator, but the velocity direction remains perpendicular to 

the direction of the Lorentz force. As a result of changing the angle from 900 to 450, the Lorentz 

magnitude will decrease to a minimum value at point C, causing a change in the electron's 

motion to make 450 with the magnetic field lines. Consequently, the electron's velocity slows 

down, and the electron loses part of its energy in the form of a photon. 

 

     After point C, due to the momentum and continuity of the electron, the electron will enter 

region 2 as shown in Figure (4). The poles of the magnet will be reversed, and thus the direction 

of the magnetic field will be reversed. According to the left-hand rule (Figure(3-b)), the velocity 

is towards the index finger and is perpendicular to the direction of the magnetic field. Therefore, 

the Lorentz magnetic force will form, which is of great value at the point where the angle is 90 

between the velocity of the electron and the Lorentz force. it causes the electron to deflect, after 

which the value of Lorentz force decreases due to a change The angle is from 90 to 45 down to 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 
 

11 

 

point D. the Lorentz magnitude will decrease to a minimum value at point D, causing a change in 

the electron's motion to make 450 with the magnetic field lines. Consequently, the electron's 

velocity slows down, and the electron loses part of its energy in the form of a photon. 

 

     After point D, due to the momentum and continuity of the electron, the poles of the magnet 

will be reversed and thus the direction of the magnetic field will be reversed electron will enter 

region 3 as shown in the Figure (4), the same simulations are repeated in Region 1. As a result, 

the electron will oscillate in a sine wave and have a wavelength λu. Moreover, a new photon is 

released due to the slowing down of the electron. Thus, a photon will be emitted in every 

undulator region (1, 2, 3, and 4).. For a certain wavelength λu of the electron, two photons will 

emit within that distance. 

    The number of the emitted photons is determined according to what the electron traverses 

from regions along the undulator (each region emits one photon) or what the electron possesses 

of one wavelength (every single wavelength through which two photons are emitted), as shown 

in Figure (4). 

 

     Since each region is similar to the other region inside the undulator, as each has similar values 

for both the magnetic field and the electron velocity, the same magnetic force will be created in 

each region according to equation (3), which notes that the reversal of the electron’s direction as 

a result of the polarity reflection of the magnets which closely resembles the electron's behavior 

(its wavelength and the loss of its energy in each region), will produce coherent photons in each 

region. These coherent photons are faster than an electron because they have the speed of light, 

so they will precede the electrons that produced them, gather in the resonator, and then leave the 

partial reflection mirror to produce the laser beam at a certain threshold. 

     The electron energy Ee value is a function of the electron path in the undulator. High-energy 

electrons have a shorter path than low-energy electrons, so coherent photons will be generated 

faster as the electron energy increases. The total energy of an electron  𝐸𝑒 represents the sum of 

the electron's kinetic energy 𝐸𝐾 and its rest energy𝐸0, as shown in the equation below:[16-19] 

 

  𝐸𝑒 = 𝐸𝐾 + 𝐸0 =  𝛾  𝑚𝑒  𝑐
2                                                (4) 

 

Where the ( 𝛾 =
1

√1−
𝑣2

𝑐2

 ) is the relativistic factor. 

𝐸𝑒 =  
𝑚𝑒  𝑐

2

√1 −
𝑣 2

𝑐2

                                                                   (5)  

 

𝑣 = 𝑐  √1 −
𝑚𝑒

2 𝑐4

𝐸𝑒
2                                                              (6)   

    

    Table (1), shows the effect of changing the electrons’ velocity 𝑣 about the Lorentz force 𝐹 

upon entering the magnetic field. As the energy 𝐸𝑒 of the electrons increases as a result of an 

increase in electrons velocity 𝑣 according to the equation (5), the values of the Lorentz force 𝐹 

will increase according to the equation (3). Figure (5) shows the sinewave behavior of the 

Lorentz force for different velocities of electrons. 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 
 

12 

 

 

 

Table 1: The Values of Lorentz force 𝐹 in the undulator. 
𝑊ℎ𝑒𝑛    𝑣 =  2.9 × 108 𝑚/𝑠 𝑊ℎ𝑒𝑛    𝑣 =  2.5 × 108 𝑚/𝑠 

PARAMETERS 𝐹 PARAMETERS 𝐹 ∅ 

3.19874e-13 𝐸𝑒    𝐽 0  ( B=0 out undulator) 1.48163e-13 𝐸𝑒    𝐽 0  ( B=0 out undulator) 90 

0.0407825 𝐵  𝑇 0  ( B=0 out undulator) 0.0407825 𝐵   𝑇 0  ( B=0 out undulator) 105 

0.02 m  uλ 0  ( B=0 out undulator) 0.02 m  uλ 0  ( B=0 out undulator) 120 

6.70054e-07 m    λ 0  ( B=0 out undulator) 0.000658896 m    λ 0  ( B=0 out undulator) 135 

1.06253e+19 w   P 0  ( B=0 out undulator) 23844.5 w   P 0  ( B=0 out undulator) 150 

 0  ( B=0 out undulator)  0  ( B=0 out undulator) 165 

0  ( B=0 out undulator) 0  ( B=0 out undulator) 180 

-4.89765e-13 -4.22212 e-13 195 

-9.46154e-13 -8.1565 e-13 210 

-1.33806e-12 -1.1535 e-12 225 

-1.63879e-12 -1.41275 e-12 240 

-1.82783e-12 -1.57571 e-12 255 

-1.89231e-12 -1.6313 e-12 270 

-1.82783e-12 -1.57571 e-12 285 

-1.63879e-12 -1.41275 e-12 300 

-1.33806e-12 -1.1535 e-12 315 

-9.46154e-13 -8.1565 e-13 330 

-4.89765e-13 -4.22212 e-13 345 

0  (sin360=0) 0  (sin360=0) 360 

4.89765e-13 4.22212e-13 375 

9.46154e-13 8.1565e-13 390 

1.33806e-12 1.1535e-12 405 

1.63879e-12 1.41275e-12 420 

1.82783e-12 1.57571e-12 435 

1.89231e-12 1.6313e-12 450 

1.82783e-12 1.57571e-12 465 

1.63879e-12 1.41275e-12 480 

1.33806e-12 1.1535e-12 495 

9.46154e-13 8.1565e-13 510 

4.89765e-13 4.22212e-13 525 

0  (sin540=0) 0  (sin540=0) 540 

 

 
Figure 5: shows the sinewave behavior of the Lorentz force for different ∅ 

    In Figure (6), it can be seen that increasing the electron's velocity and approaching the speed 

of light will lead to an exponential increase in the electron's energy Ee and the power P of the 

output laser according to equation (5) and equation (2), respectively. While the increase is linear 

-3.00E-12

-2.00E-12

-1.00E-12

0.00E+00

1.00E-12

2.00E-12

3.00E-12

0 100 200 300 400 500 600

F
 (

N
) 

(degree) ∅

E1

E2



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 
 

13 

 

and gradual in Lorentz's force according to Equation (3). Finally, the wavelength of the output 

laser will decrease linearly and gradually according to equation (1).  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 6: The effects of increasing the electron's velocity. 

 

    Now, the analytical methods will be illustrated below to show the effect of the Lorentz force 

in FEL. The beam-radiation interaction in the undulator is described by classical physics while in 

quantum physics effects are expected to be small. Consider an ultra-relativistic electron 

traversing in the undulator magnetic field described by the equation below.  

 

𝐵 = 𝐵0 sin( 𝑘𝑢𝑥)                                                                 (7) 

Where (𝑘𝑢 =
2𝜋 

𝜆𝑢
)   and 𝐵0 is the magnetic pole field. When the velocity 𝑣 is assumed to be equal 

to the speed of light c, and based on equations (3,4 and 7), we obtain 

 

𝛾  𝑚𝑒  
𝑑𝑣

𝑑𝑡
 ≈ −𝑒(𝑣 × 𝐵)   

    Yields two coupled equations 

 

 
𝑑𝑣𝑥

𝑑𝑡
 ≈ −  

𝑒𝑣𝑧𝐵𝑦

𝛾 𝑚𝑒 
     𝑎𝑛𝑑    

𝑑𝑣𝑧

𝑑𝑡
 ≈ −  

𝑒𝑣𝑥𝐵𝑦

𝛾 𝑚𝑒 
                                (8) 

0.00E+00

5.00E-13

1.00E-12

1.50E-12

2.00E-12

2.50E-12

3.00E-12

3.50E-12

2.40E+08 2.80E+08 3.20E+08

E
e

  
 J

v   m/s

Ee

0.00E+00

5.00E-13

1.00E-12

1.50E-12

2.00E-12

2.50E-12

2.40E+08 2.80E+08 3.20E+08

F
  
N

v  m/s

F

-500

0

500

1000

1500

2000

2500

3000

2.00E+08 2.50E+08 3.00E+08 3.50E+08

P
  

W

v    m/s

P

-0.0005

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

2.00E+08 2.50E+08 3.00E+08 3.50E+08

λ
m

v    m/s

λ



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 
 

14 

 

 

      Integrating equation (8) and (𝑣𝑧  ≈ 𝑣 = 𝛽𝑐 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡) , ( 𝑣𝑥  ≪  𝑣𝑧) , (
𝑑𝑣𝑧

𝑑𝑡
 ≈ 0) , leads to 

obtaining the transverse velocity and the solution for 𝑥(𝑡), 𝑧(𝑡) . 

𝑣𝑥 ≈  
𝑒𝐵0

𝛾 𝑚𝑒 𝑘𝑢 
cos( 𝑘𝑢𝑧)                             (9) 

 

𝑥(𝑡) ≈  
𝑒𝐵0

𝛾 𝑚𝑒 𝛽𝑐𝑘𝑢
2

 
sin( 𝑘𝑢𝛽𝑐𝑡)                (10) 

 

 𝑧(𝑡) ≈  𝛽𝑐𝑡                                                   (11) 

    From the equations above, an important dimensionless undulator parameter K has been 

obtained, which is equal to 

 

𝐾 =  
𝑒𝐵0

𝑚𝑒𝑐 𝑘𝑢 
=

𝑒𝐵0 𝜆𝑢
2𝜋 𝑚𝑒𝑐 𝑘𝑢 

= 0.934 𝐵0(𝑇𝑒𝑠𝑙𝑎 𝑢𝑛. ) 𝜆𝑢(𝑐𝑚 𝑢𝑛. )  (12) 

 

    in Figure (7), it can be seen that increasing the magnetic field will lead to an exponential 

increase in the wavelength of the output laser and a linear increase in the undulator parameter K 

according to equation (1) and equation (12), respectively. While an exponential decrease in the 

power P of the output laser according to Equation is observed (2).  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 7: The effects of increasing the magnetic field. 

0

1

2

3

4

5

6

7

0 2 4

K
(T

m
) 

B (T)

B vs K

0

50

100

150

200

250

0 2 4

𝝀
𝑭
𝑬
𝑳

(𝒏
𝒎

) 

B (T)

B vs λ FEL 

0

50000

100000

150000

200000

250000

0 2 4

P
 𝑭
𝑬
𝑳

(W
) 

B (T)

B vs P FEL



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 
 

15 

 

 

 

4. Conclusions 
     From the simulation results obtained, it can be concluded that the Lorentz force is responsible 

for generating the laser beam in the free electron laser. Coherent photons are emitted as the 

electron gains an oscillatory movement that accelerates and decelerates it according to the 

direction of the magnetic field. Lorentz's power can be controlled by the velocity of the electrons 

released from the electron launcher and the resulting change in the wavelength power of the 

output laser. 

 

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