Analyses of vertical atmospheric upward laser propagation for laser optical communication [1, 2], satellite laser ranging (SLR) [3, 4], laser weaponry [5], and laser energy transmission [6] are routinely conducted in various fields. When a laser beam propagates through the atmosphere, atmospheric turbulence causes beam scattering and distortion [7, 8]. This leads to a degradation in signal quality, with energy dispersing or pointing errors occurring before reaching the target [9]. For these reasons, it is crucial to predict and analyze the propagation characteristics of laser beams.

Figure 1 illustrates laser beam propagation from ground to space through atmospheric disturbance. The most significant atmospheric disturbance is concentrated from the ground up to an altitude of 20 km, especially near the ground, known as the ground layer. Figure 2 demonstrates the changes or degradation in laser intensity distribution due to atmospheric disturbance, including the beam wander effect [Fig. 2(a)], where the beam shifts randomly, and scintillation [Fig. 2(b)], where the beam’s intensity fluctuates. These effects are shown for long exposure [Fig. 2(c)] and short exposure [Fig. 2(d)], and indicate different impacts based on exposure time.

Figure 1. Upward laser atmospheric propagation from ground position to space.

Figure 2. Example of initial laser intensity distribution and laser intensity distribution after laser atmospheric propagation. The effects that appear after propagation are the (a) beam wander effect and (b) scintillation effect. Depending on the exposure time, it is expressed as (c) long exposure and (d) short exposure.

Research to minimize beam distortion caused by atmospheric turbulence is also essential to enhance the performance of laser communications, SLR, laser weaponry, and energy transmission systems. Recent studies on vertical atmospheric upward laser propagation have been conducted in various fields, including self-focusing analysis for high-power lasers [10–12] and range bias estimation for ranging purposes [13, 14].

In Korea, several studies focus on ground-to-space lasers, primarily at the Geochang SLR Observatory. Despite this, no research has reported on vertical laser beam propagation under the specific atmospheric conditions of this observatory. Our research uses a 100-watt-class laser for experiments, including laser guide star (LGS) adaptive optics (AO), to address this gap.

This paper explores how various laser beam parameters affect ground-to-space laser propagation up to 100 km. Using data from vertical atmospheric disturbance profiles at the Observatory, we analyze key parameters such as laser wavelength, beam size (diameter), beam jitter, and beam quality (M ²). The goal is to predict laser performance at higher altitudes and optimize laser applications amid atmospheric disturbances.

The structure of this paper is as follows: Section 2 provides an overview of the numerical analysis used. Section 3 presents the several ranges of parameters for analysis and describes the method. Section 4 analyzes the results and suggests optimal parameters for each application field. Finally, Section 5 summarizes the findings.

2.1. Numerical Analysis of Laser Propagation

To understand the complex interactions between the characteristics of the laser beam and atmospheric state variations, we initially explain the theory constituting the numerical analysis.

First, the scalar wave equation, expressed as Eq. (1), is used to describe the propagation of the laser beam:

Note that this equation is represented using the complex wave function ψ, the wavenumber k, and the refractive index variation δε. ${\nabla }_{\perp }^2$ denotes the transverse Laplacian operator, which explains the essential wave characteristics related to laser beam propagation.

Second, the energy balance equation is used to understand the distribution and variation of energy in the atmosphere, expressed as Eq. (2):

This equation includes information on temperature variation T₁, wind velocity $\overrightarrow{V}$, thermal conductivity, density ρ₀, specific heat capacity C_p, absorption coefficient α_a, and laser intensity I_p. It quantifies the absorption and transmission processes of laser energy in the atmosphere, which explains the impact of temperature and wind velocity changes on energy distribution.

Third, the refractive index variation equation represents changes in the refractive index due to thermal expansion, which is crucial to explaining the refractive index changes of the laser beam. The refractive index variation is primarily determined by temperature and is expressed as Eq. (3) with the temperature-dependent refractive index change coefficient ∂n/∂T.

δε_TB represents the refractive index changes due to temperature and pressure and illustrates the changes in the optical path of the laser beam resulting from these variations.

2.2. Phase Numerical Analysis

In laser atmospheric propagation, the atmosphere exerts several effects on the laser. The refractive index changes occur due to atmospheric turbulence, alter the phase and intensity of the laser beam [15, 16]. Additionally, aerosols and gas molecules in the atmosphere cause energy loss through laser absorption and scattering. These effects are expressed in terms of phase.

The numerical analysis of phase is composed of random screens simulating atmospheric turbulence and nonlinear phase screens reflecting distortions due to thermal variations. These phase screens represent changes in the refractive index at each position where the beam passes, and reflect effects such as atmospheric turbulence. The equations used are based on the Von-Karman spectrum and the frozen layer theory and define the phase changes of the beam.

The phase screen generation equation based on the von Kármán spectrum is given in Eq. (4):

Φ_n represents the turbulence spectrum density at spatial frequencies $k_x^2$ + $k_y^2$, $C_n^2$ is the turbulence strength coefficient, and $k_0^2$ is the wavenumber of the inner scale. Equation (5) is used to represent the delay time caused by atmospheric turbulence using the Greenwood time delay:

where r₀ is the Fried parameter, representing the beam diameter affected by atmospheric turbulence, and v_effective signifies the wind speed.

The phase screens constructed in this manner are used for each section to calculate how the beam’s phase changes. Each phase screen models the phase change in a specific section of the atmosphere, and by applying these sequentially, the entire beam path is reconstructed. As shown in Fig. 3, multiple phase screens are arranged sequentially to visually demonstrate the gradual phase changes as the beam passes through each screen. This method allows for precise simulation of the laser’s atmospheric propagation path in the numerical analysis.

Figure 3. Arrangement of random phase screens and nonlinear phase screens in the numerical analysis.

Simulation of laser beam intensity distribution over distances up to 100 km in the presence of atmospheric turbulence using numerical analysis is shown in Fig. 4. As the propagation distance increases, the laser beam’s intensity distribution becomes increasingly irregular, and the beam’s central fluctuation grows larger. At 0 km, the initial beam exhibits strong energy concentration at the center. However, as the distance progresses to 10 km and 20 km, the beam becomes increasingly distorted and forms multiple intensity peaks. At 50 km and 100 km, the intensity distribution becomes even more irregular and the beam’s coherence is significantly reduced. This demonstrates that the beam’s performance gradually deteriorates with distance due to atmospheric turbulence. These findings highlight the necessity for adaptive optics and correction techniques in long-distance laser communication and aviation systems.

Figure 4. Example of a beam intensity distribution simulation over varying distances in the presence of atmospheric turbulence.

3.1. Application of Domestic Atmospheric Data

The measurement of atmospheric characteristics uses various instruments such as slope detection and ranging (SLODAR) and differential image motion monitor (DIMM) [17, 18]. In 2023, Kongju National University in South Korea developed a SLODAR system at the Geochang SLR Observatory to measure domestic atmospheric characteristics, and it is currently conducting atmospheric measurements [18]. The Geochang Observatory is dedicated to satellite laser ranging using a 1.0 m SLR telescope and is located at the summit of Mt. Gamak, at an altitude of 952 m with coordinates 35°35’24.0”N 127°55’12.0”E [18, 19]. SLODAR can measure $C_n^2$ and the Fried parameter at different altitudes of the vertical atmosphere using a crossed-beam method based on binary stars. The measurements were conducted at a wavelength of 500 nm using the narrow mode with a small separation angle between the binary stars to capture the atmospheric characteristics at higher altitudes. Figure 5 shows the location of the Geochang SLR Observatory and the installed SLODAR system, and Fig. 6 presents the refractive index structure constant measured over nine days in May 2024, represented as best, median, and worst conditions.

Figure 5. Location and development image of SLODAR: (a) Location of Geochang SLR Observatory, (b) Image of the developed SLODAR [17]. SLODAR, slope detection and ranging; SLR, satellite laser ranging.

Figure 6. Cn2 by altitude measured with SLODAR equipment at Geochang SLR Observatory (for nine days in May 2024). SLODAR, slope detection and ranging; SLR, satellite laser ranging.

Additionally, the Fried parameter can be expressed as an integral of $C_n^2$ over altitude, as shown in the following equation. Table 1 presents the Fried parameter calculated through the integration of $C_n^2$ at different altitudes [20, 21]. The best and worst conditions refer to the top 10% and bottom 10% of the measured data, respectively.

TABLE 1. The best, median, and worst Cn2 values measured at Geochang SLODAR Observatory are calculated as r₀.

Cn2	Best	Median	Worst
r0 (cm)	10.89	7.33	4.13

3.2. Numerical Analysis Parameters and Ranges

Short-wavelength lasers are more susceptible to scattering in the atmosphere, but may be advantageous for specific applications due to their higher energy. In contrast, long-wavelength lasers experience less absorption and scattering in the atmosphere, allowing them to propagate over longer distances. The radius and jitter of the laser beam directly affect the beam’s focusing ability and targeting precision. A smaller radius and lower jitter ensure high precision but also increase the system’s complexity and cost. The M ² value represents the beam quality; a lower M ² value indicates a higher quality beam, which requires greater precision in manufacturing and maintenance. The comprehensive range of parameters for this study includes the following as tabulated in Table 2.

TABLE 2. Laser parameter range used in numerical analysis.

Parameter	Range of Variation
Wavelength (μm)	0.532–1.550
Initial Laser Radius (cm)	2–30
Jitter (μrad)	1–10
M 2	1–5

4.1. Short- and Long-exposure Final Beam Radius

In applications such as LGS and SLR, the radius of the laser beam after atmospheric propagation is a crucial factor affecting performance and accuracy. A smaller final beam radius leads to a more accurate artificial star in LGS applications and improves the precision of distance measurements to satellites in SLR applications. We defined the final beam radius as the distance from the beam’s center to the point where the beam intensity in the plane perpendicular to the direction of propagation decreases to 1/e (approximately 36.8%) of its maximum value, at the target plane, i.e., 100 km altitude.

This analysis was conducted for two distinct cases: short-exposure beams, which capture instantaneous turbulence effects, and long-exposure beams, which average the effects of turbulence over time. For our study, we applied 0.02 seconds for the short-exposure and 10 seconds for the long-exposure. Figure 7 illustrates the variations in long-exposure and short-exposure beam radius sizes based on parameter changes as in Table 2.

Figure 7. Long exposure and short exposure radius size according to parameter changes: (a) Wavelength changes from 532 to 1,550 nm, (b) laser radius changes from 2 to 30 cm, (c) M ² changes from 1 to 5, and (d) jitter 1 varies up to ~10 μrad.

The first thing to mention is that the long-exposure beam is approximately two to three times larger than the short-exposure beam, except under jitter conditions. The next thing to notice is that the final beam radius, especially long-exposure, increases relatively linearly with the beam wavelength, the beam jitter, and the beam quality (M ²) but not with the initial beam radius. Regarding the final beam radius, the minimum or optimum is achieved at the initial beam radius of around 17 cm, as in Fig. 7(b). This indicates that the beam’s ability to maintain focus diminishes beyond this radius, which leads to larger beam spread on the target surface.

In Fig. 7(c), a higher M ² value results in a larger divergence angle of the laser beam and causes the laser energy reaching the target to spread over a wider area. This increase in divergence reduces the beam’s intensity on the target, thereby increasing both the long-exposure and short-exposure radii. Figure 7(d) shows that, unlike other parameters, the gap between the long-exposure radius and the short-exposure radius gradually widens as jitter increases. This suggests that jitter has less impact on the intensity distribution without any overall beam shift. It also shows that jitter primarily affects the minimum output beam size under the numerical analysis conditions.

Overall, the long-exposure beam radius ranged from 40 to 120 cm, while the short-exposure beam radius varied between 15 and 45 cm. These variations provide insights into how different parameters influence the propagation characteristics of laser beams, and help optimize laser system performance in varying atmospheric conditions.

4.2. Pointing Errors

As the laser propagates through the atmosphere, the beam center randomly deviates due to laser jitter and the beam wander effect caused by atmospheric turbulence. This deviation is a critical analysis factor for applications that require the laser beam to reach precise locations, such as satellites or space probes.

In this study, the extent of beam center displacement is expressed as the pointing error. To calculate the pointing error, Eqs. (7) and (8) were used to determine the shift in the beam’s central axis in μrad and the results were converted to the root mean square (RMS) for expression. A larger pointing error value indicates a higher likelihood that the laser beam will deviate from the target point or fail to aim precisely at the target.

x_c and y_c represent the weighted centroids of x and y, respectively, and S(x, y) denotes the weighting value between x and y. Σ_x and Σ_y represent the summation over the coordinates in each direction.

By analyzing the pointing error, it was observed that deviation in the beam center increases with higher jitter values and larger M ² values, while wavelength and radius also significantly affect pointing accuracy. The results, depicted in Fig. 8, show how each parameter influences the pointing error of the laser beam, which emphasizes the importance of controlling these parameters for precise targeting in various applications.

Figure 8. Pointing error according to parameter changes: (a) Wavelength changes from 532 to 1,550 nm, (b) laser radius changes from 2 to 30 cm, (c) M ² changes from 1 to 5, and (d) jitter changes from 1 to 10 μrad.

The laser radius showed a linear decrease, whereas the pointing error increased linearly with the values of parameters other than the laser radius. In Figs. 8(a) to 8(c), the pointing error varied between 1.5 and 1.9 μrad. However, jitter, a parameter that directly affects beam center fluctuation, exhibited a pointing error ranging between 1.5 and 3.0 μrad.

4.3. Scintillation

Scintillation is one of the effects caused by a laser beam passing through the atmosphere, and results in fluctuations in the distribution of laser intensity. In this study, scintillation is analyzed using Eq. (9), expressed as the scintillation index. A higher scintillation index indicates greater intensity fluctuations occurring as the laser beam traverses the atmosphere.

<I> represents the average intensity of the laser beam, and <I ²> denotes the mean square of the intensity over time or space. This index indicates how much the intensity deviates from the average value. Figure 9 illustrates the scintillation index according to parameter changes as in Table 2.

Figure 9. Scintillation index according to parameter changes: (a) Wavelength changes from 532 to 1,550 nm, (b) laser radius changes from 2 to 30 cm, (c) M ² changes from 1 to 5, and (d) jitter changes from 1 to 10 μrad.

As the value of jitter increased, the scintillation index also increased linearly. In contrast, the scintillation index decreased linearly with the other parameter values. Overall, the scintillation index varied between 1.0 and 2.2 μrad.

4.4. Influence Analysis Based on Parameters

The influence of each parameter on vertical laser atmospheric propagation at 100 km was analyzed considering domestic atmospheric conditions, as shown in Fig. 10. This analysis illustrates the effects of parameters on the final beam radius (long exposure, short exposure), pointing errors, and scintillation index based on the data presented in Figs. 7–9.

Figure 10. Effect depending on the amount of change in each parameter.

In the analysis of the long-exposure radius, jitter emerged as the most significant influencing factor. Jitter directly reflects the pointing error of the laser beam and has an approximately 6.7 times greater impact compared to wavelength. This indicates that in laser systems where precise targeting is essential, the impact of jitter must be taken into account.

Conversely, for short-exposure radius, the M ² value has the most substantial influence. M ², which determines the beam quality and intensity distribution, has an approximately 11 times greater impact than jitter. Furthermore, jitter also has the greatest impact on pointing error, with about 17 times greater influence than the M ² value. Therefore, compensation for jitter is crucial in the design of laser systems.

Additionally, in a performance influence analysis, it is essential to minimize the long-exposure or short-exposure radii at the point where the output beam radius is minimal. In fields where uniform intensity distribution is critical, adjusting the M ² value or wavelength to larger values can be beneficial. However, it is important to note that a larger wavelength results in an increased beam radius. In summary, the performance of vertical laser atmospheric propagation, considering domestic atmospheric conditions, is significantly influenced by jitter and the M ² value, which each play a critical role in different laser applications.

Table 3 categorizes the key variables and their impact factors in various laser applications. For laser guide stars, the long-exposure beam diameter is the most critical for creating artificial stars, and jitter is the most sensitive factor. In optical communications and energy transmission, maintaining accurate laser pointing and energy quality makes pointing error and scintillation index the most important, with jitter and M ² being the most sensitive factors. For SLR, pointing error is crucial for precise laser targeting, and jitter has the most significant impact. Overall, jitter was found to be the most influential factor in laser propagation applications, highlighting the need for further research on the various influences of atmospheric laser propagation.

TABLE 3. Categorization of key variables and their impact factors in various laser applications.

Application Field	Key Variables	Maximum Impact Factor
Laser Guide Star	Long Exposure Diameter	Jitter
Laser Optical Communication	Pointing Error, Scintillation	Jitter, M 2
Laser Energy Transmission	Pointing Error, Scintillation	Jitter, M 2
Laser Ranging	Pointing Error	Jitter

This study analyzed vertical upward laser propagation at 100 km, considering domestic atmospheric conditions. Using the refractive index structure constant ($C_n^2$) values measured at the Geochang SLR Observatory, we examined the impact of key parameters such as laser jitter, wavelength, and beam quality (M ²) on laser propagation characteristics.

The analysis revealed that laser jitter is the most influential factor affecting the long-exposure radius, with an approximately 6.7 times greater impact compared to the least influential parameter, wavelength. In contrast, for the short-exposure radius, beam quality (M ²) was found to be the most significant factor, with an influence approximately 11 times greater than that of jitter. Furthermore, in terms of pointing error, jitter again emerged as the most impactful parameter, with about 17 times greater influence compared to beam quality (M ²). Additionally, for the scintillation index, beam quality (M ²) was identified as the most influential factor, with an influence approximately 16.8 times greater than that of jitter. The key variables and impact factors in various laser applications were categorized, and it was found that jitter is the most significant factor in laser propagation.

Defense Rapid Acquisition Technology Research Institute (DRATRI) grant funded by the Defense Acquisition Program Administration (DAPA) (Grant no. UC200013D).

The authors declare no conflicts of interest.

Data underlying the results presented in this paper are not publicly available at this time, but may be obtained from the authors upon reasonable request.