124 

© 2025 The Author(s). Published by College of Education for Pure Science (Ibn Al-Haitham), 

University of Baghdad. This is an open-access article distributed under the terms of the Creative Commons 

Attribution 4.0 International License 

   

Simulate and Analyze the Effect of the Magnetic System on the Electron 

Oscillatory Motion and the Formation of the Free Electron Laser Beam 

 

Jaafar H. Sabri1*  and Thair A. Khalil Al-Aish 2  
1,2 Department of Physics, College of Education for Pure Science (Ibn Al-Haitham), University of 

Baghdad, Baghdad, Iraq. 

*Corresponding author.  

Received :24 April 2024                   Accepted :10 September 2024                 Published: 20 April 2025 

doi.org/10.30526/38.2.3994 

 
Abstract   

     The free electron laser (FEL) represents a significant advance over conventional laser 

systems by allowing a wide range of operating wavelengths without changing the active laser 

medium or undergoing extensive redesign. Central to this ability is the Wiggler component, 

which influences many of the properties of the emitting laser through its wavelength (𝜆𝑢) This 

study investigates the influence of (𝜆𝑢) on the FEL's performance metrics, including magnetic 

field strength, photon wavelength, and laser power, by utilizing MATLAB R2023a for 

simulation. The study employs a set of equations to explore how variations in (𝜆𝑢) affect these 

parameters. The undulator's role, a critical component of the FEL, is analyzed for its ability to 

convert high-speed electron kinetic energy into coherent light by manipulating electron 

trajectories through Lorentz force interactions. Results indicate a direct proportionality 

between (𝜆𝑢) and both the generated magnetic field and output photon wavelengths, suggesting 

that (𝜆𝑢) is a pivotal factor in optimizing FEL performance. The findings highlight potential 

enhancements in laser power and stability that could benefit a range of applications, from 

medical to military. This paper not only reaffirms the critical role of (𝜆𝑢) in the operational 

efficiency of FEL systems but also guides future design strategies for advanced laser 

technologies. 

Keywords: Free Electron Laser, Wiggler, Laser power, Lorentz force, Undulator.   

 

1. Introduction 

     The invention of the free electron laser system fifty years ago this decade caused a 

tremendous revolution in human life and its general applications, including medical and 

military applications. (1,2). Laser technologies have been developed that outperform traditional 

laser systems with unique advantages. This technology is called free electron laser (FEL) (3), 

as it is characterized by its ability to generate a wide spectrum of laser wavelengths without the 

need to change the material of the laser active medium or redesign the entire system (4). In a 

free electron laser system, the active medium consists of particles of negatively charged 

electrons traveling at very high speeds approaching the speed of light after being supplied with 

energy from a voltage source (5). The active medium will not be a solid, liquid metal, or 

semiconductor as in conventional laser systems (6,7). These negatively charged electrons line 

https://creativecommons.org/licenses/by/4.0/
https://creativecommons.org/licenses/by/4.0/
https://orcid.org/0009-0008-5397-5051
mailto:Jaafar.Sabri2204p@ihcoedu.uobaghdad.edu.iq
https://orcid.org/0000-0003-2479-5840
mailto:thair.ak.i@ihcoedu.uobaghdad.edu.iq


IHJPAS. 2025,38(2) 

125 
 

up with each other while traveling as a stream in the form of a beam, and this beam or stream 

of electrons passes through two rows of magnets with certain geometric dimensions called an 

undulator (8,9). where in the undulator the magnets are arranged periodically, that is, once a 

south pole is followed by a north pole, and so it continues. That is, it is done, in an alternating 

and opposite way, by generating a regular periodic magnetic field (10 -12). This alternating 

magnetic field generated by the undulator obstructs and inhibits the movement of the electron 

beam passing through it, forcing it to move in an oscillatory manner and bending the path of 

its movement, where a force called the Lorentz force will arise (13). The wavelength of Wiggler 

(𝜆𝑢) is one of the most important factors that directly affect the magnetic field, which in turn 

affects the electronic beam in particular, the wavelength of the laser output, and the power of 

the laser as a whole in general (1,12) By conducting a literature survey and reviewing previous 

research that devoted its attention and calculations to studying the free electron laser system 

and working to improve its performance The literature survey and review revealed that the 

effect of the wavelength of Wiggler (𝜆𝑢) on the free-electron laser system had not been studied 

extensively from various aspects, whether its effect on the magnetic, the wavelength of the 

laser output, or the total energy of the free-electron laser system. Despite its great importance 

in influencing the generation of coherent photons and the oscillatory movement of the electron 

(14). The focus of this work was to analyze the effect of the wavelength of Wiggler (𝜆𝑢) on the 

path of the electron and the generation of photons inside the laser. 

 

2. Materials and Methods 

     To obtain a wide and variable range of wavelengths in the electromagnetic spectrum, many 

parameters can be studied in the free-electron laser system. This wide spectrum can be used in 

many scientific, medical, and military applications. Without the need to replace the system as 

a whole or change the type of active medium (15,16), but only by making some modifications 

in the geometric design of the free-electron laser system (17), because the free-electron laser is 

characterized by the ability to generate a wide range of wavelengths, extending from radio 

wavelengths to X-rays (1-3). In this paper, certain geometric changes were made in the 

undulator magnets so that from these changes, we obtain variable values for the wavelength of 

Wiggler (𝜆𝑢). Making these changes leads to a change in the laser output in terms of wavelength 

and laser output power. Making these changes in the free electron laser is one of the important 

advantages that add important characteristics to the system, including wide stability, short 

wavelengths, high brightness, pulsed laser, short pulses, and high energy (18). The working 

principle of the free electron laser is based on the idea of the presence of an active medium, 

which is a current beam of electrons that travel at high speeds approaching the speed of light 

and then passes through a magnetic field (9). These electrons are accelerated through an 

electron accelerator, and these electrons pass through the magnetic field generated by the 

undulator (2). The main components of the free electron laser system are the following: The 

electron gun, which fires negative electron particles and is equipped with a high voltage source, 

is a system that works to provide the electrons with the necessary kinetic energy; an electron 

accelerator is a system that increases the speed of electrons to very high speeds. The undulator 

is a system consisting of magnets arranged in two rows that generate a regular magnetic field 

through which the electron beam passes. Its working principle is to convert part of the kinetic 

energy of electrons traveling at high speeds into coherent electromagnetic radiation. A 

resonator is a system of mirrors that amplifies the laser beam (6, 19-21). 

 



IHJPAS. 2025,38(2) 

126 
 

 
Figure 1. Sketch producing a pair of photons in an undulator in difference (𝜆𝑢) (1). 

 

After using equations found in the literature and scientific references and performing some 

calculations, the following was calculated: The effect of the wavelength of Wiggler (𝜆𝑢) on the 

magnetic field (B), and the relationship between them was found and drawn graphically. The 

oscillatory parameter (k) and Lorentz factor (γ) were also calculated for energy (500 MeV), 

which is equivalent to (8 ∗ 10−10) in joules. The wavelength of the photon (𝜆) was also 

calculated at the same energy mentioned, and the relationship between the wavelength of 

Wiggler (𝜆𝑢) and the wavelength of the resulting photon (𝜆) was drawn graphically; the initial 

energy (P0), photon energy (Ep), number of photons (𝑁ph), and gain were also calculated. gain-

length (LG), laser power (𝑃𝑧), for the mentioned energy (500 MeV). 

𝐵 = 4.22 𝑒𝑥𝑝(−
𝑔𝑢

𝜆𝑢
(5.08 + 1.54

𝑔𝑢

𝜆𝑢
))                                                                  (1) 

Where:(B) is the magnetic field, (𝑔𝑢) is undulator gap, (𝜆𝑢) wavelength of the wiggler (22). 

𝐾 = 93.3𝜆𝑢𝛽𝑢                                                                                                       (2) 

𝛾 =
𝐸𝑒

𝑚𝑒𝑐2
                                                                                                                  (3) 

𝜆 = 𝜆𝑢(
1

2𝛾2
+

𝑘2

4𝛾2
)                                                                                                       (4) 

Where: (k) wiggler parameter, (𝛾) Lorentz-factor, (Ee) energy of electrons particles, (λ) 

wavelength of laser (22,23)  

𝑃0 =
1

9

𝜒2𝑐𝐸𝑒

𝜆
                                                                                                        (5) 

Where: (𝑃0) initial power, (𝜒) scaling parameter takes the value (10-2 or 10-3) (24,25). 

𝐸𝑝 = ℎ
𝑐

𝜆
                                                                                                                    (6) 

Where: (𝐸𝑝ℎ) photon energy (22,25). 



IHJPAS. 2025,38(2) 

127 
 

𝑁𝑃𝐻 =
𝜒𝐸𝑒

𝐸𝑝ℎ
                                                                                                                     (7) 

Where: (𝑁𝑃𝐻) number of photons (23,26). 

𝐿𝐺 =
𝜆𝑢

4𝜋√3𝜒
                                                                                                                     (8) 

Where: (𝐿𝐺) gain-length (21,27). 

𝑃𝑧 = 𝑁𝑃𝐻𝑃0 𝑒𝑥𝑝(𝑧 𝐿𝐺)⁄                                                                                              (9) 

Where: (PZ) The laser power, (z) length of undulator (1). 

 

3.  Results and Discussion  

     In this paper, the executive program (FEL) was created and designed using (MATLAB 

R2023a (9.14.0.2206163)) as shown in Figure 2. The mathematical interface of the executive 

program shown in Figure 2. consists of several parameters and equations to simulate the 

requirements of the calculations that were completed for this paper in terms of the wavelength 

of Wiggler (𝜆𝑢). During the study of this paper, some factors affecting the wavelength 

formation of the photons generated in the free electron laser system were studied, such as the 

magnetic field (B), as well as Several other important parameters were also calculated, which 

are (Lorentz factor (γ), photons wavelength (λ), initial power (P0), photon energy (Ep), number 

of photons (𝑁𝑝ℎ), gain-length (𝐿𝐺), The laser power (𝑃𝑧), out power from the mirror (𝑃𝑚)).The 

undulator is considered an important and main part of the free electron laser system, as it 

consists of a group of small magnets, the number of which is always even facing each other in 

two rows, working to generate a regular magnetic field. These pieces of small magnets are 

arranged among themselves periodically, once at the north pole and then at the next pole. 

Southern and so on the most important benefit envisaged for the undulator system is the 

generation of regular magnetic field lines measured in the Tesla unit It works to slow down the 

very high speeds of the electron beam particles coming from the accelerator after they have 

been supplied with high energy through a high current voltage source whose range ranges from 

one megaelectron volt to five hundred megaelectron volts. 

 

Figure 2. Shows the simulation program to analyze the effect of wavelength of Wiggler (𝜆𝑢) on the 

output laser. 



IHJPAS. 2025,38(2) 

128 
 

When the electrons reach very high speeds close to the speed of light and then enter the 

undulator, the regular magnetic field will obstruct the path of these particles and transform their 

path from a linear path to a curved path (29). This resulting curved path consists of peaks and 

troughs, and the distance between every two identical points along the path is called the 

wavelength of Wiggler (𝜆𝑢), and it is repeated along the undulator. Where two photons are 

generated, one at the top of the wave and the other at the bottom, as shown in Figure 1. In this 

paper, we will study the effect of changing the wavelength of Wiggler (𝜆𝑢), which is the 

distance between any two identical points along the path of electrons inside the undulator. 

Where calculations and simulations have proven that the magnetic field (B) is directly 

proportional to the wavelength of Wiggler (𝜆𝑢) and there is a positive relationship between the 

laser wavelengths (𝜆) to the wavelength of Wiggler (𝜆𝑢) where the effect of the wavelength of 

Wiggler (𝜆𝑢) on the magnetic field was calculated through equation (1) when the undulator gap 

𝑔𝑢= 0.03 and energy of electron particles 𝐸𝐵𝑀 = 8 ∗ 10−10 joule and Lorentz-factor γ = 

9771.46 The calculation results from the simulation are recorded in Table 1. To calculate the 

effect of changing the wavelength of Wiggler (𝜆𝑢) on the magnetic field (B) and the laser 

wavelengths produced in the free electron laser (λ), this was done using equations (1-4) Where 

the effect of changing the wavelength of Wiggler (𝜆𝑢) was studied at the energy of the electron 

particles (8 ∗ 10−10) joules, which is equivalent in the unit of electron volts (500 MeV). The 

value of the Wiggler parameter (k) was also calculated through equation (2). Where it was 

considered that the mass of the electron is 𝑚𝑒=9.10938356*10−31 (kg) and the typical speed 

of light is C=299,792,458 (m⁄s). The values resulting from the calculations and simulations 

have been recorded in Table 1. 

 

Table 1. Magnetic field (B) and Wiggler parameter(k) and Wavelength values (λ) resulting from changing the 

wavelength of Wiggler (𝜆𝑢) at 500 MeV. 

𝒈𝒖 = 𝟎. 𝟎𝟑 & 𝑬𝑩𝑴 = 𝟖 ∗ 𝟏𝟎−𝟏𝟎𝑱 & γ= 9771.46 

 Wavelength 

of undulator 

(𝝀𝒖) in (m) 

Magnetic field 

(B) in (T) 

Wiggler parameter 

(k) in Tm 

Laser wavelength 

(𝝀) in nm 

1 0.01 9.71277e-13 9.06201e-13 0.0523662 

2 0.012 8.50527e-10 9.5225e-10 0.0628394 

3 0.014 6.70848e-08 8.76262e-08 0.0733127 

4 0.016 1.37202e-06 2.04815e-06 0.0837859 

5 0.018 1.2316e-05 2.06835e-05 0.0942592 

6 0.02 6.47378e-05 0.000120801 0.104732 

7 0.022 0.000236143 0.000484707 0.115206 

8 0.024 0.00066454 0.00148804 0.125679 

9 0.026 0.00154613 0.0037506 0.136153 

10 0.028 0.00311702 0.0081429 0.14663 

11 0.03 0.00562708 0.0157502 0.157118 

 



IHJPAS. 2025,38(2) 

129 
 

  

Figure 3. Plot of Change in a magnetic field (B) 

resulting from changing the wavelength of Wiggler (𝜆𝑢) 

at 500 MeV. 

Figure 4. Plot of Change in laser wavelength values 

(𝜆) resulting from changing the wavelength of 

Wiggler (𝜆𝑢) at 500 MeV. 

 

Figure 3 shows that there is a direct relationship between the wavelength of Wiggler (𝜆𝑢) and 

the magnetic field (B), which are linked together within equation (1), where the wavelength of 

Wiggler represents the distance between any two identical points along the curved path of the 

electron beam. Which is exposed to the magnetic field by the undulator system, where the 

magnetic field begins at (9.71277e-13) in Tesla when the wavelength of Wiggler (𝜆𝑢) is equal 

to (0.01 m) and ends at (0.00562708 T) if the wavelength of Wiggler (𝜆𝑢) is equal to (0.03 m), 

Through the behavior of the curve, the physical meaning of the curve can be explained by 

saying that there is a direct interaction between both the negative particle field and the 

undulator magnetic field. While Figure 4 shows that there is a linear relationship between the 

wavelength of Wiggler (𝜆𝑢) and laser wavelength values (λ), which are linked together within 

equation (4), where the simulation showed that any increase in the wavelength of Wiggler will 

be offset by an increase in laser wavelength. values (λ) and the opposite are true, meaning that 

whenever the wavelength of Wiggler decreases, there will be a decrease in laser wavelength 

values (λ). Therefore, the wavelength of Wiggler is considered an important factor in obtaining 

a wide spectrum of wavelengths for the laser output, as can be seen from the diagram that laser 

wavelength values (λ) will be (0.0523662 nm) when the wavelength of Wiggler equals (0.01 

m) while laser wavelength values (λ) will be equal to (0.157118 nm) when the wavelength of 

Wiggler equals (0.03 m), Through the behavior of the curve, the physical meaning of the curve 

can be explained by saying that every increase in the wavelength of Wiggler (𝜆𝑢) It means an 

increase in the amount of obstruction of electron particles at 500 MeV Therefore, there will be 

an increase in laser wavelength values (λ). In order to calculate the effect of changing the 

wavelength of Wiggler (𝜆𝑢) on the initial power (𝑃0) in the free electron laser, this was done 

using equation (5) At the energy studied (500 MeV), where the values and constants have been 

substituted, the (χ) scaling parameter takes a value of (0.001).To calculate the laser power (𝑃𝑧) 

is done through equation (9) where photon energy (𝐸𝑝ℎ) was calculated through equation (6), 

and the number of photons (𝑁𝑝ℎ) was also calculated through equation (7). Likewise, gain 

length (LG) was calculated through equation (8). length of undulator (𝐿𝐷) was considered (2m). 

The values resulting from the calculations and simulations have been recorded in Table 2. 

 

 

 

 

 

 

-1.00E+10

0.00E+00

1.00E+10

2.00E+10

3.00E+10

4.00E+10

5.00E+10

6.00E+10

0 0.01 0.02 0.03 0.04

(B
) 

in
 (

T)

(𝝀_𝒖) in (m)

4

6

8

10

12

14

16

18

0.009 0.014 0.019 0.024 0.029 0.034

(𝝀
)*

1
0

^-
9

(𝝀_𝒖) in (m)



IHJPAS. 2025,38(2) 

130 
 

Table 2. Values of initial power (𝑃0) and photon energy (Ep) values of (𝑁𝑝ℎ) number of photons resulting from     

changing the wavelength of Wiggler (𝜆𝑢) of energy 500 MeV. 

𝝌 = 𝟎. 𝟎𝟎𝟏  &  𝑬𝑩𝑴 = 8 ∗ 10−10𝐽 & 𝑳𝑫 = 𝟐𝒎 

 Laser 

wavelength 

(𝝀)in nm 

Initial 

power  

(𝑷𝟎) in (w) 

Photon 

energy 

(𝑬𝒑𝒉) In FJ 

NO. 

photon 

(𝑵𝒑𝒉) 

gain-

length 

(𝑳𝑮) 

The laser 

power (𝑷𝒛) 

In J 

1 0.0523662 508.882 3.79337 210.894 0.459674 8.32257e+06 

2 0.0628394 424.069 3.16115 253.072 0.551609 4.0302e+06 

3 0.0733127 363.487 2.70955 295.252 0.643543 2.40093e+06 

4 0.0837859 318.051 2.37086 337.43 0.735478 1.62804e+06 

5 0.0942592 282.712 2.10743 379.609 0.827413 1.2035e+06 

6 0.104732 254.442 1.89669 421.787 0.919348 945083 

7 0.115206 231.309 1.72426 463.967 1.01128 775497 

8 0.125679 212.034 1.58057 506.147 1.10322 657664 

9 0.136153 195.723 1.45898 548.328 1.19515 572063 

10 0.14663 181.738 1.35473 590.524 1.28709 507610 

11 0.157118 169.606 1.2643 632.761 1.37902 457657 

 

 

  
Figure 5-a. Plot of initial power (𝑃0) resulting from 

changing the wavelength of Wiggler (𝜆𝑢) at 500 

MeV. 

Figure 5-b. Plot of Photon Energy (𝐸𝑝ℎ) resulting from 

changing the wavelength of Wiggler (𝜆𝑢) at 500 MeV. 

 

  

Figure 5-c. Plot of values of a number of photons 

(𝑁𝑝ℎ) resulting from changing the wavelength of 

Wiggler (𝜆𝑢) at 500 MeV. 

Figure 5-d. Plot of values of gain-length (𝐿𝐺) 

resulting from changing the wavelength of Wiggler 

(𝜆𝑢) at 500 MeV. 

 

0

100

200

300

400

500

600

0 0.01 0.02 0.03 0.04

(𝑷
_𝟎

) 
in

 (
w

)

(𝝀_𝒖) in (m)

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.01 0.02 0.03 0.04

(𝑬
_𝒑
𝒉

) 
*1

0
^-

1
5

(𝝀_𝒖) in (m)

0

100

200

300

400

500

600

700

0 0.01 0.02 0.03 0.04

(𝑵
_𝒑
𝒉

)

(𝝀_𝒖) in (m)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.01 0.02 0.03 0.04

(𝑳
_𝑮

)

(𝝀_𝒖) in (m)



IHJPAS. 2025,38(2) 

131 
 

 
Figure 5-e. Plot of the laser power (𝑃𝑧) resulting from changing the wavelength of Wiggler (𝜆𝑢) at 500 MeV. 

 

Figures (5-a) and (5-b) show through simulation calculations that there is an inverse 

relationship between the wavelength of Wiggler (𝜆𝑢) with both initial power (P0) and Photon 

Energy (𝐸ph) at the energy studied. Whereas, the initial power is the integrated shot noise 

incoherently radiated into the dominant growing mode by the electrons passing through the 

first gain length, Likewise, Photon Energy (𝐸ph) started at (210.894 femtojoules) and decreased 

to (1.2643 femtojoules) when the wavelength of Wiggler (𝜆𝑢) was increased. This proves that 

the wavelength of Wiggler (𝜆𝑢) plays an important role in both important factors. Figures (5-

c) and (5-d) show through simulation calculations that there is a direct relationship between 

the wavelength of Wiggler (𝜆𝑢) with both the number of photons (𝑁ph) and the gain-length 

(LG), and that any change in the amount of Gain-length will affect laser power (𝑃𝑧), because 

gain-length (LG) is one of the requirements for calculating equation (9). As a result, this means 

that the wavelength of Wiggler (𝜆𝑢) affects them greatly. Figure (5-e) Simulation calculations 

found that there is a negative relationship between the wavelength of Wiggler (𝜆𝑢) with the 

laser power (𝑃𝑧), The resulting values were recorded through equation (9) based on calculating 

all the requirements of the aforementioned equation at the same energy studied.As for changing 

the wavelength of Wiggler (𝜆𝑢), it depends mainly on the geometry and size of the magnet 

pieces that make up the undulator system, as whenever the poles are changed along the path of 

the electron, the direction of the top or bottom of the path changes, and all of this is subject to 

the influence of the Lorentz force. 

 

4. Conclusion 

     Through the simulations that were conducted using scientific equations related to the subject 

of the free electron laser mentioned in the references and sources, and through the simulation 

program that was designed, the calculations that were conducted showed that there is a large 

and direct effect of the wavelength of Wiggler (𝜆𝑢), on each of the laser output wavelengths. 

The laser's power also plays a significant role. As the electron beam flows into the undulator 

system, many options can be used to obtain a change in the behavior of the laser system as a 

whole. This paper states that the magnetic field (B), Laser Wavelength (𝝀), initial power (P0), 

Photon Energy (𝐸ph), number of photons (𝑁ph), the gain length (LG), and the laser power (𝑃𝑧) 

are affected when there is any change in the wavelength of Wiggler (𝜆𝑢), and that there is a 

positive proportionality between the wavelength of Wiggler (𝜆𝑢) with the magnetic field (B), 

Laser Wavelength (𝝀), number of photons (𝑁ph) and the gain-length (LG), while there is 

negative proportionality with initial power (P0), Photon Energy (𝐸ph), and the laser power (𝑃𝑧) 

at the energy studied in this paper of 500 MeV. 

 

0

2

4

6

8

10

0 0.01 0.02 0.03 0.04

(𝑷
_𝒛

)*
1

0
^6

(𝝀_𝒖) in (m)



IHJPAS. 2025,38(2) 

132 
 

Acknowledgment 

     My thanks and appreciation to the College of Education for Pure Science, Ibn Al-Haytham, 

University of Baghdad, for helping to complete this work by providing scientific facilities from 

the Physics Department to help support the research project. 

 

Conflict of Interest  

     The authors declare that they have no conflicts of interest. 

 

Funding 

     None. 

 

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