IHJPAS. 36 (3) 2023 

124 

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

 

  

 

 

Simulation and Analysis the Attenuation Effect of Atmospheric Layers on a 

Laser Beam Within the Visible Range 

   

 
*Corresponding Author: mohammed1985kamal@gmail.com 

 

 

 

 

 

Abstract 

The power and the size of the final spot of the laser beam reaching the target are very important 

requirements in most of the laser applications and fields such as medical, military, and scientific, 

so studying laser propagation in the atmosphere is a very important topic. The propagation of the 

laser beam through the atmosphere is subject to several attenuation processes that deplete the 

power and expand the beam. Through the simulation results of the free electron laser within the 

visible region of the electromagnetic spectrum (400-700nm), it was found that the attenuation 

increases with decreasing wavelength. Laser propagation in the presence of rain and snow leads to 

a very large loss of power compared to propagation in normal weather conditions free of rain and 

snow. Atmospheric turbulence depends largely on changes in temperature, so the turbulence 

decreases with altitude from sea level, which makes laser work at high altitudes, such as the 

stratosphere, a good option with better results. 

Keywords: Free Electron Laser, Attenuation Coefficient, M2, Gaussian Beam, Atmospheric 

Turbulence   

1. Introduction 

In recent years, lasers have been used in many aspects of our lives, including communications, 

defense, industry, and scientific research. Studying the variables that influence the laser beam's 

power, beam diameter, and other characteristics is therefore crucial and necessary. The 

propagation of laser beams and the final spot size are important factors in most laser applications. 

doi.org/10.30526/36.3.3093 

Article history: Received 31 October 2022, Accepted 21 November 2022, Published in July 2023. 

 

 

 

 

 Mohammed Kamal Saleh*  

Department of Physics, College of Education for Pure 

Sciences Ibn Al-Haitham, University of Baghdad, 

Baghdad, Iraq. 

 

mohammed1985kamal@gmail.com 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Ibn Al-Haitham Journal for Pure and Applied Sciences 

Journal homepage: jih.uobaghdad.edu.iq 

 

 Thair Abdulkareem Khalil Al-Aish  

Department of Physics, College of Education for Pure 

Sciences Ibn Al-Haitham, University of Baghdad, 

Baghdad, Iraq. 

 

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

 

 

 

https://creativecommons.org/licenses/by/4.0/
mailto:mohammed1985kamal@gmail.com
mailto:mohammed1985kamal@gmail.com
mailto:mohammed1985kamal@gmail.com
mailto:thair.ak.i@ihcoedu.uobaghdad.edu.iq
mailto:thair.ak.i@ihcoedu.uobaghdad.edu.iq


IHJPAS. 36 (3) 2023 
 

125 
 

The propagation of laser beams in atmospheric layers is subject to attenuation operations 

(scattering, absorption, rain, snow, etc.) and atmospheric turbulence. The final power of the laser 

and beam divergence are affected by these phenomena [1,3]. The primary determinants of the 

performance quality of light effects are the features of the light source's beam and how they change 

along the optical axis of propagation and throughout the beam. The transverse mode (TEM00), 

which is similar to the output light for single-mode optical devices, defines the output of many 

laser devices built and used to produce or emit this mode (TEM00). The ideal (TEM00) beam is 

distinguished by the symmetry and revolution of the optical axis, in addition to having the 

maximum intensity in the center. Its intensity is consistent with that of the Gaussian beam [2]. 

In this study, laser beam propagation in the atmosphere will be examined, and the effects of 

attenuation operations on laser beams in the visible range (400–700 nm) will be calculated in terms 

of power loss and beam divergence. 

 

2.Theory           

In this study, the laser is obtained within the visible range through the free electron laser system 

(FEL). An FEL consists of three important components: (an electron accelerator, an undulator, and 

an optical resonator. FEL has important and unique features, as it can obtain a laser of any 

wavelength within the range from IR to X-rays. The following equations are used to calculate the 

required wavelength [4-5]: 

 𝜆 =
𝜆𝑤
𝛾 2

(
1

2
+

𝐾2

4
) (1) 

 

 𝐾 =
𝑒 𝜆𝑤 𝛽𝑤
2𝜋𝑚𝑒 𝑐

= 93.22 × 𝜆𝑤 × 𝛽𝑤 (2) 

Where e: charge of electron, 𝑚𝑒 : mass of electron,  λw the undulator period length , 𝛾  the 

relativistic factor , 𝛽𝑤  Intensity of magnetic field c: velocity of light and  𝐸𝑏𝑒𝑎𝑚  the energy of  the 

beam. The magnetic field of an undulator 𝐵𝑤 can be calculated by[6]: 

 
𝐵𝑤 = 4.22 𝑒𝑥𝑝 [−

𝑔𝑤
𝜆𝑤

(5.08 + 1.54
𝑔𝑤
𝜆𝑤

)] 
(3) 

Where 𝑔𝑤 is the distance between the poles of the undulator. 

The output power 𝑃𝑤 from the undulator can be calculated by[6][7]: 

 𝑃𝑤 = 𝑃𝑖𝑛  𝑁 𝑒𝑥𝑝(
𝐿𝑤
𝐿𝑔

) (4) 

Where 𝑃𝑖𝑛 is initial power, 𝑁 the number of coherent photon per electron, 𝐿𝑤  length of undulator and 𝐿𝑔  the gain 

length   

Finally, the following equation determines the total power of the free electron laser that emerges 

from the optical resonator[7],[9]: 

 𝑃 = 𝑃𝑤  𝑒𝑥𝑝(−𝜎𝑑) (5) 

 P: is the total power of laser, Pw is the output power from the undulator, σ is attenuation coefficient  

and d is the path length[10]. 



IHJPAS. 36 (3) 2023 
 

126 
 

Note: Saturation power 𝑃𝑠𝑎𝑡  given in this relation 𝑃𝑠𝑎𝑡 =  𝜒 (𝐸𝑏𝑒𝑎𝑚 × 𝐼𝑏𝑒𝑎𝑚) =

0.001(1250 × 106 × 10000) = 12.5𝐺𝑤). Where, 𝜒 is Pierce parameter and 𝐼𝑏𝑒𝑎𝑚 is the current 

of electron beam. 

When the laser beam exits the aperture of the mirror and propagates in the atmosphere, it 

undergoes a deviation from the axis of propagation according to diffraction theory. As a result, 

depending on the atmosphere's density, length of the propagation axis, and interactions with 

components of the atmosphere, The final spot size of the laser changes due to wave propagation. 

In this paper, the propagation of a Gaussian TEM00 laser beam in the atmosphere within the visible 

spectrum (400–700 nm) will be investigated and simulated. The basic radius of a laser beam WO 

can be calculated by[11]–[13] 

 
𝑊𝑜 = [

𝐿 𝜆

𝜋
]

1/2

  
(6) 

The final radius of central Gaussian beam in Near-Mid Field given by[14] can be as follows: 

 

𝑊 = 𝑊𝑜  [[1 + (
𝑑 𝜆

𝜋𝑊𝑜
2

)
2

]]

1/2

 

(7) 

In Far field, the final radius of laser beam is given by[11]: 

 𝑊 = 𝑊𝑜 +
𝑑 𝜆

𝜋𝑊𝑜
 (8) 

According to diffraction theory, as the propagation axis length increases, so does the 

diameter of the laser beam. The laser beam divergence can be calculated using the following 

equation[2], [10]: 

 
𝜃 =

𝑀2𝜆

𝜋𝑊𝑜
 

(9) 

The Rayleigh range is an important parameter in determining the propagation field of the 

laser beam in the near, mid, and far field, and the Rayleigh range value can be calculated using the 

following equation[2], [11]: 

 𝑍 =
𝜋𝑊𝑜

2

𝜆
 (10) 

The quality factor, which is the ratio between the theoretical Gaussian beam and the actual, is 

utilized to determine the performance quality of any optical device. The quality factor can be 

calculated by[11-12]: 

 
𝑀2 =

𝑊

𝑊𝑜
 (11) 

The atmosphere is composed of a variety of gases, some of which are present in varying 

amounts; the main gases are hydrogen, nitrogen, oxygen, and so on, which are drawn to the earth 

by gravity. The density of the atmosphere varies with altitude, temperature, and pressure. The 

atmosphere is divided into five layers based on pressure and temperature. Interactions with 

atmospheric particles and gases influence laser beam propagation in the atmosphere. Laser beam 

power is lost as a result of these interactions, and beam divergence occurs[15-16]. 

The propagation of laser beams through the atmosphere is a topic that has been of 

considerable theoretical and practical interest for a long time. When a laser beam propagates 

through the atmosphere, it subject to attenuation operations such as absorption, scattering, rainfall, 



IHJPAS. 36 (3) 2023 
 

127 
 

snowfall, and turbulence. The laser beam interacts with atmospheric components such as gases, 

particles, dust, smoke, and so on, and the amount of absorption and type of scattering depend on 

the relationship between the size of the atmospheric components and the wavelength of the laser 

beam. As a result of these interactions, the final laser power is reduced. The following equation is 

used to calculate the amount of absorption and scattering attenuation[9], [15]: 

 𝜎𝑎𝑏+𝑠𝑐 =
3.912

𝑑
[
550

𝜆
]

𝑞

 (12) 

Where, d altitude and λ laser wavelength in (nm). 

Attenuation also occurs when it snows and rains. Since the size of the snowflake is larger 

than the raindrops, the loss in power of the laser transmitted during snowfall is more significant 

compared to the beam transmitted during rain. Thus, it represents a significant obstacle in the path 

of the laser beam. The amount of attenuation coefficient due to snowfall can be calculated by the 

equation[11-12]: 

 𝜎𝑠𝑛 = 𝑞𝑆
𝑢 (13) 

Where S is the snow rate in millimeters per hour and (q, u) represents the value of dry and wet 

snow. The attenuation coefficient due to rainfall can be calculated by the following equation[11], 

[12]: 

 𝜎𝑟𝑎 = 1.076 𝑆
0.67 (14) 

The movement of air between the layers of the atmosphere and close to the Earth's surface 

causes large fluctuations in air density and refractive index, which result in turbulence in the air. 

These fluctuations are caused by the change in temperature and pressure with altitude. 

Atmospheric turbulence greatly affects the propagation of light waves. The amount and strength 

of atmospheric turbulence decrease with altitude above sea level. The refractive index is a function 

of temperature and pressure, and can be calculated by[11], [15]: 

 𝑛 = 1 + [
77.6 × 10−6 × 𝑃

𝑇
] [1 +

7530

𝜆2
] (15) 

𝜆 is laser wavelength in (nm). The amount of turbulence can be calculated from the structural 

refractive index parameter  𝐶𝑛
2 with the following equation[11], [15], [17]: 

 
𝐶𝑛

2 = 〈(𝑛2 − 𝑛1)
2〉  × 𝑑

−
2
3 

(16) 

Where, 𝑛1: the refractive index of first area, 𝑛2 : the refractive index of second area and d the 

distance between two areas. 

The Fried parameter is a crucial variable when examining how turbulence in the atmosphere affects 

the propagation of a Gaussian laser beam. It represents the circular diameter over the beam that 

preserves coherence in the propagation path length. Strong turbulence is indicated by the Fried 

parameter's highest value, and vice versa. The Fried parameter can be calculated by[11], [15][18]: 

 
𝑟𝑜 = 0.33 × 𝜆

6
5  × (𝑑

3
5  × 𝐶𝑛

2)
−

3
5
 

(17) 

3. Results and Discussion 

The results for this paper were obtained using Matlab software in version R2018b, where a 

program was developed to simulate the spread of a Gaussian beam from a free electron laser in the 

visible range. To generate a visible laser, a free electron laser was simulated using the designed 

implementation simulation program (400-700nm). A variety of factors, including the undulator 



IHJPAS. 36 (3) 2023 
 

128 
 

period, relativistic factor, static magnetic field, undulator gap, and others, have an effect on the 

laser's wavelength, according to the simulation. The table below depicts a portion of the simulation 

of a free electron laser for obtaining the required laser. 

Table 1: Shows the simulation of Free Electron Laser. 

E (PJ) Relativistic ɣ λ FEL (nm) PFEL (GW) 

40 487.864 760.822 312.928 

42 512.258 690.086 345.003 

44 536.651 628.777 378.645 

46 561.044 575.29 413.847 

48 585.437 528.348 450.619 

50 609.83 486.926 488.95 

52 634.224 450.189 528.85 

54 658.617 417.46 570.314 

56 683.01 388.174 613.342 

58 707.403 361.865 657.933 

60 731.797 338.142 657.933 

 

  

Figure 1. The relation between E beam  and  ɣ. Figure 2. The relation between λ FEL and E beam. 

The amount of electron beam energy that affects the wavelength of the free electron laser system 

is shown in Table 1 and Figures 1 and 2. According to equation 1, the laser wavelength is 

inversely proportional to the relativistic factor. The relativistic factor increases as the energy of the 

electron beam increases, implying that the laser wavelength is inversely proportional to the energy 

of the electron beam. 

  

Figure 3. The relation between λ FEL and λw Figure 4. The relation between λ FEL and K. 

0

100

200

300

400

500

600

700

800

40 42 44 46 48 50 52 54 56 58 60

R
e

la
ti

vi
st

ic
 ɣ

 

E beam (PJ)

0

100

200

300

400

500

600

700

800

40 42 44 46 48 50 52 54 56 58 60

λ
F

E
L 

(n
m

)

E beam (PJ)

0

100

200

300

400

500

600

700

800

0
.0

4

0
.0

3
9

0
.0

3
8

0
.0

3
7

0
.0

3
6

0
.0

3
5

0
.0

3
4

0
.0

3
3

0
.0

3
2

0
.0

3
1

0
.0

3

λ
F

E
L 

(n
m

)

λw(m)

0

100

200

300

400

500

600

700

800

0 2 4 6

λ
F

E
L 

(n
m

)

K



IHJPAS. 36 (3) 2023 
 

129 
 

The undulator period is the second factor in calculating the FEL. Figure 3 shows that the FEL 

increases as the undulator period increases and vice versa. Where the electron is accelerated and 

decelerated as it moves between the poles of the undulator. When it moves up and down, it emits 

energy in the form of a photon. According to Eq 1, reducing the undulator period results in a 

decrease in the wavelength FEL generated by the undulator as well as an increase in power. The 

undulator parameter is also an important factor in calculating the wavelength of the laser. The 

wavelength is directly proportional to the magnitude of the undulator parameter, i.e., the greater 

the value of the undulator parameter, the greater the wavelength, and vice versa. The other factor 

that affects the amount of laser wavelength is the undulator parameter, which is directly 

proportional to the wavelength according to Equation 1, where the higher the value of the undulator 

parameter, the greater the amount of wavelength generated from the free electron laser system and 

vice versa, as shown in Figure 4. 

  

Figure 5. The relation between g w and λFEL Figure 6. The relation between g w and β . 

According to Equation 3, the strength of the static magnetic field (β) is proportional to the 

magnitude of the gap between the two poles, where the magnetic field strength decreases (i.e., 

inversely proportional) with an increase in the gap of the undulator, as shown in Figure 5, and 

since the value of the undulator parameter (K) is proportional to the strength of the static magnetic 

field(β) according to Equation 2, it turns out to behave similarly (i.e., inversely proportional 

According to Eq. 1, the laser wavelength is proportional to the magnitude of the undulator gap; the 

wavelength decreases as the undulator gap decreases. 

The propagation of the laser beam through the atmosphere is critical, and according to diffraction 

theory, the propagation laser beam will be subject to deviation in its path, resulting in beam 

expansion due to a variety of factors such as density, attenuation processes, and atmospheric 

turbulence. Eq. 8 calculates the radius of the laser beam produced by the mirror aperture. With a 

10m optical resonator length and λFEL (690.086 nm), the basic radius of the laser is equal to 

0.00148247 m, as illustrated in Table 2. 

 

 

 

 

 

 

 

0

100

200

300

400

500

600

700

800

0
.0

1

0
.0

1
1

0
.0

1
2

0
.0

1
3

0
.0

1
4

0
.0

1
5

0
.0

1
6

0
.0

1
7

0
.0

1
8

0
.0

1
9

0
.0

2

λ
F

E
L 

(n
m

)

g w (m)

0

0.2

0.4

0.6

0.8

1

1.2

0
.0

1

0
.0

1
1

0
.0

1
2

0
.0

1
3

0
.0

1
4

0
.0

1
5

0
.0

1
6

0
.0

1
7

0
.0

1
8

0
.0

1
9

0
.0

2

β

g w(m)



IHJPAS. 36 (3) 2023 
 

130 
 

 

 

 Table 2: shows the Gaussian beam propagation in altitude (1-11km).  

 

 

 

  
Figure 7. The relation between M2 and height d. Figure 8.The relation between W (m) and height d. 

Figures 7 and 8 show the divergence of the fundamental laser beam as well as the quality factor 

values of the laser beam traveling through the atmosphere, in accordance with Table 2. In this 

example, the expanded laser beam radius has deviated from the ideal value of the basic beam radius 

because it grows as the propagation path length (height) does. Additionally, because the quality 

factor depends on how much of the expanded radius is present, as determined by Eq. 13, its value 

has deviated from the ideal value. The value of the quality factor indicates (M2=101) an increase 

in the base laser beams radius at altitude (d=1km). The amount of quality factor present is 

determined by the extended beam radius, which is determined by the Rayleigh field. The greater 

the Rayleigh field, the greater the distance the beam can travel without deviating from the ideal 

value, and the less power is lost.  

 

 

 

0

500

1000

1500

2000

2500

3000

0 10 20 30

M
2

d (km)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 10 20 30

W
 (

m
)

d (km)

λ nm initial wo (m) d (Km) final w (m) M2 Divergence 

690.086 0.00148247 Mean sea level 0.00148247 1 0.000148248 

690.086 0.00148247 1 0.14973 101 0.014973 

690.086 0.00148247 2 0.297978 201 0.0297979 

690.086 0.00148247 3 0.446225 301 0.0446227 

690.086 0.00148247 4 0.594473 401 0.0594476 

690.086 0.00148247 5 0.74272 501 0.0742723 

690.086 0.00148247 6 0.890968 601 0.0890971 

690.086 0.00148247 7 1.03922 701 0.103922 

690.086 0.00148247 8 1.18746 801 0.118746 

690.086 0.00148247 9 1.33571 901 0.133572 

690.086 0.00148247 10 1.48396 1001 0.148397 

690.086 0.00148247 11 1.63221 1101 0.163222 



IHJPAS. 36 (3) 2023 
 

131 
 

  

  
Divergence  Power  

Figure 9. Shows The divergence and power of gaussian beam of FELin 2D and 3D. 

Figure 9 depicts the divergence cone of the free electrode laser's Gaussian beam as well as the 

amount of power. The divergence grows with height, and the cone contains approximately 87% 

energy. To achieve high directional beams, short wavelengths with a large spot size must be used. 

As the wavelengths get longer, the divergence gets smaller. The power, on the other hand, is largely 

determined by the amount of frequency and the Rayleigh field, as it is directly proportional to the 

frequency and inversely proportional to the wavelength. When it comes to the Rayleigh field, it is 

highly dependent on the distance traveled, in the case of constant distance, as well as the amount 

of the Rayleigh field. Figure 9 represents the divergence and power of a Gaussian beam with a 

frequency of (434241GHz) i.e. (690.086nm) in 2 and 3 dimensions.  

  

Figure 10. The relation between T and height  d . Figure 11. The relation between P  and height d. 

Figures (10 and 11) show the correlation between altitude above sea level and temperature and 

pressure. As the altitude increases above sea level, the temperature in the troposphere, the first 

layer of the atmosphere, will decrease by a fixed amount known as the decreasing constant, which 

equals 6.5 kelvin per kilometer (temperature is inversely proportional to altitude). The relationship 

is inverse because pressure decreases as altitude increases. As altitude above sea level rises, the 

atmospheric pressure decreases because a drop in temperature causes a drop in the kinetic energy 

0

50

100

150

200

250

300

350

0 10 20 30

T
 (

K
)

d (Km)

0

200

400

600

800

1000

1200

0 10 20 30

P
 (

m
b

a
r)

d (Km)



IHJPAS. 36 (3) 2023 
 

132 
 

of the atmosphere's gaseous constituents. As you ascend above sea level, density will behave 

similarly because it is influenced by temperature and pressure. 

Table 3.  The attenuation coefficient of various visible wavelength with intial power (1000MW) 

d (km) 
𝜆 = 690.086 𝑛𝑚 𝜆 = 528.348 𝑛𝑚  𝜆 = 417.46 𝑛𝑚 

σ(sc+ab) (1/km) power (MW) σ(sc+ab) (1/km) final P (MW) σ(sc+ab) (1/km) final P (MW) 

1 3.42573 32.5255 4.005 18.2243 4.59675 10.0846 

2 1.6548 36.5308 2.01476 17.7829 2.39675 8.28341 

3 1.07681 39.5405 1.34894 17.4779 1.64554 7.17882 

4 0.792199 42.0542 1.01516 17.238 1.26339 6.38656 

5 0.623526 44.2619 0.814474 17.037 1.0309 5.77337 

6 0.51227 46.2534 0.680439 16.863 0.874057 5.27729 

7 0.4161 54.3288 0.588811 16.2173 0.799784 3.70346 

8 0.364088 54.3286 0.51521 16.2172 0.699811 3.70346 

9 0.323634 54.3285 0.457964 16.2173 0.622055 3.70343 

10 0.29127 54.3288 0.412168 16.2172 0.559849 3.70345 

11 0.264791 54.3288 0.374698 16.2173 0.508954 3.70344 

Table 3 displays the attenuation coefficients for three wavelengths of visible light (the study area) 

due to absorption and scattering. It is evident that the attenuation coefficient increases as the laser 

wavelength decreases (inverse proportion), according to equation 12. Furthermore, it is evident 

from Table 3 that the amount of loss power of a laser varies depending on the laser wavelength, 

which is related to the amount of attenuation. It has been observed that the final power will 

decrease with each of the three wavelengths as they get shorter. 

Table 4.  The final power after attenuation for laser beam 

d (Km) q 

Without Snow & Rain Attenuation With Snow & Rain Attenuation 

σ(Sc+ab) (1/km) final P (MW) σ tot(1/km) Final P (W) 

1 0.585 3.42573 32.5255 6.213433 2002360 

2 0.737 1.6548 36.5308 4.442503 138450 

3 0.8437 1.07681 39.5405 3.864513 9225.58 

4 0.9287 0.792199 42.0542 3.579902 603.283 

5 1.00034 0.623526 44.2619 3.411229 39.2231 

6 1.063 0.51227 46.2534 3.299973 2.51795 

7 1.3 0.4161 54.3288 3.203803 0.182075 

8 1.3 0.364088 54.3286 3.151791 0.0112088 

9 1.3 0.323634 54.3285 3.111337 0.000690017 

10 1.3 0.29127 54.3288 3.078973 0.00004248 

11 1.3 0.264791 54.3288 3.052494 0.000002615 

 

  

Figure 12. The relation between (σ) and altitude d 

Without  Snow & Rain Attenuation 

Figure 13. The relation between power and d (km) 

Without  Snow & Rain Attenuation 

0

0.5

1

1.5

2

2.5

3

3.5

4

0 10 20 30

σ
(S

c+
a

b
) 

1
/k

m

d (km)
0

10

20

30

40

50

60

0 10 20 30

F
in

a
l 

p
o

w
e

r 
(M

w
)

d (km)



IHJPAS. 36 (3) 2023 
 

133 
 

  
Figure 14. The relation between (σ) and altitude d 

With Snow & Rain Attenuation. 

Figure 15. The relation between power and d (km) 

With Snow & Rain Attenuation. 

Table 4 and Figures (12-15) show the magnitude of the loss power of the laser beam (𝜆𝐹𝐸𝐿 =

690.086𝑛𝑚   𝑎𝑛𝑑 𝑃𝑜 = 1000𝑀𝑊) with and without snow and rain attenuation. From Figures 12 

and 13, we see that the amount of loss in the final laser power that reaches the target depends 

largely on the amount of the attenuation coefficient, which in turn depends on the wavelength of 

the laser beam. As we explained earlier and in Figures 13 and 14, we see that the amount of loss 

due to attenuation without adding rain and snow attenuation will gradually decreases with 

increasing altitude, as the greatest loss is in the first 1 km due to the high density and abundance 

of particles, such as dust grains, dust, etc., that increase the amount of attenuation. At a height of 

8 km or more, the attenuation coefficient begins to stabilize at very close values, as in Figures 14 

and 15. The attenuation coefficient and power loss with the presence of rain and snow attenuation 

show that the amount of power coming out of the device, which is 1000 MW, has greatly 

diminished with altitude. This is due to the fact that the attenuation of snow is greater than the 

attenuation of rain, which in turn is greater than the attenuation of absorption and scattering 

because the size of the snowflake is very large and causes a large attenuation of the laser power. 

Table 5.  Show the change of main parameters of atmosphere turbulence with altitude 

d (km) Temp K P mbar n Cn2 (m -2/3)×10-12  ro  (m) 

1 281.65 898.661 1.000251513 5.670571469 0.006209001 

2 275.15 794.801 1.0002277 4.861187818 0.006810082 

3 268.65 700.883 1.000205652 4.15180548 0.0074861 

4 262.15 616.162 1.000185276 3.532029414 0.008248656 

5 255.65 539.933 1.000166483 2.99263363 0.009110992 

6 249.15 471.527 1.000149183 2.524397765 0.010090278 

7 242.65 410.317 1.000133295 2.120009731 0.011204567 

8 236.15 355.706 1.000118735 1.771712976 0.012478483 

9 229.65 307.137 1.000105424 1.473187291 0.013939366 

10 223.15 264.084 1.000093287 1.218339357 0.015622007 

11 216.65 226.055 1.000082249 1.438072696 0.014142601 

12 216.65 193.096 1.000070257 1.04925766 0.017087106 

13 216.65 164.943 1.000060014 0.765580238 0.020644453 

14 216.65 140.895 1.000051264 0.55867665 0.024940289 

15 216.65 120.352 1.000043789 0.407604096 0.030133854 

16 216.65 102.805 1.000037405 0.297417499 0.036406413 

17 216.65 87.8162 1.000031951 0.217015223 0.043984907 

18 216.65 75.0127 1.000027293 0.158348272 0.05314101 

19 216.65 64.0759 1.000023314 0.115539666 0.064203564 

20 216.65 54.7337 1.000019915 0.088903992 0.075135476 

21 217.65 46.7536 1.000016933 0.064259327 0.09129271 

22 218.65 39.937 1.000014398 0.046449308 0.110920099 

23 219.65 34.1142 1.000012243 0.033575819 0.134766464 

24 220.65 29.1404 1.00001041 0.024271411 0.163734722 

25 221.65 24.8918 1.000008852 0.017555733 0.198859533 

 

0

1

2

3

4

5

6

7

0 5 10 15

σ
 t

o
ta

l 
(1

/k
m

)

d (km)

0

0.5

1

1.5

2

2.5

0 5 10 15

F
in

a
l 

P
 (

M
W

)

d (km)



IHJPAS. 36 (3) 2023 
 

134 
 

  

Figure 16. The relation between Cn2  and  d (km) in 

troposphere  

Figure 17. The relation between Cn2  and d (km)  

above troposphere  

  

Figure 18. The relation between  ro(m)  and d (km) 

in troposphere  

Figure 19. The relation between  ro(m)  and d (km) 

above  troposphere 

  

Figure 20. The relation between  n  and d (km) in 

troposphere 

Figure 21. The relation between  n  and d (km) 

above troposphere 

Table 5 shows the main parameters that go into calculating atmospheric turbulence. 

Figures 16 and 17 show that the value of the structural refractive index 𝐶𝑛
2 decreases with 

increasing altitude from sea level due to the decrease in temperature with altitude, where 

fluctuations in temperature are a major cause of turbulence, as shown in equation 16. Figures 18 

and 19 show the values of a Fried parameter that increases with height inversely proportional to 

the structural refractive index, as shown in equation 17. The amount of air refractive index depends 

0

1

2

3

4

5

6

0 5 10 15

C
n

2
 (

m
 -

2
/3

)×
1

0
-1

2

d (km)

0

0.2

0.4

0.6

0.8

1

1.2

12 13 14 15 16 17 18 19 20 21 22 23 24 25

C
n

2
 (

m
 -

2
/3

)×
1

0
-1

2

d (km)

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0 5 10 15

ro
 (

m
)

d (km)

0

0.05

0.1

0.15

0.2

0.25

12 13 14 15 16 17 18 19 20 21 22 23 24 25

ro
 (

m
)

d (km)

1.00008

1.0001

1.00012

1.00014

1.00016

1.00018

1.0002

1.00022

1.00024

1.00026

0 5 10 15

n

d (km)

0.99997

0.99998

0.99999

1

1.00001

1.00002

1.00003

1.00004

1.00005

1.00006

1.00007

1.00008

12 13 14 15 16 17 18 19 20 21 22 23 24 25

n

d (km)



IHJPAS. 36 (3) 2023 
 

135 
 

largely on the amount of temperature, and therefore the refractive index will decrease with 

increasing height, which is linearly proportional to the temperature according to Equation 15. As 

shown in Figures 20 and 21, the structural refractive index is linearly proportional to pressure. 

From the above, it proves to us that temperature is the main factor in everything that occurs during 

turbulence in the atmosphere. 

 

4.Conclusions 

Through the simulation results in this study, we conclude that the laser can be produced in the 

free electron laser system at any desired wavelength by changing and manipulating the basic 

factors involved in calculating the wavelength, such as the energy of the beam, the undulator 

period, the undulator gap, etc. Also, the attenuation coefficient increases with short wavelengths, 

so, blue light is more susceptible to attenuation compared to red light. The greatest amount of 

absorption is near the surface of the earth due to the high density of air as well as the presence of 

particles and suspended objects such as dust, which increases the attenuation and thus the increase 

in the loss in the final power. The amount of atmospheric turbulence depends largely on the 

temperature fluctuation, so the atmosphere is more turbulent near the sea surface, and this 

turbulence decreases with an increase in altitude. The radius of the laser beam increases with the 

increase in height, which leads to an increase in the beam divergence. 

 

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