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Abstract
In this study, we have deduced the generalized scintillation index of the plane wave and plane wave structure function based on the modified non-Kolmogorov power spectrum of atmospheric refractive index with the spectrum power index α. In addition, we have analyzed the fluctuation of atmospheric density due to thermal blooming. Based on the interaction of six beams in thermal blooming, the thermal phase screening and intensity distribution are simulated. Under the influence of atmospheric turbulence and thermal blooming, the six-beam combination is then simulated numerically to obtain the equivalent radius with long exposure (RL), power in the bucket (PIB), Strehl ratio (SR), and peak value of intensity (Ip). Results show that PIB, Ip, and SR of the pulse beam combination decrease with an increase in α; however, RL operates in reverse mode and the short pulse durations reduce the thermal blooming. Moreover, laser of short duration cannot generate high ring energy on the target.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In recent years, the discussion on marine disputes has attracted the attention of researchers [1,2,3]. Some countries utilize the laser coherent or incoherent beam combining as directed weapons to preserve their rights on the ocean. Therefore, it is important to understand the propagation characteristics of high energy laser in the marine atmosphere. Under the influence of atmospheric turbulence and thermal blooming, high energy laser propagates in the marine atmosphere. The atmospheric turbulence causes beam wander, spot spreading, and intensity scintillation of laser [4]. This severely weakens the energy focusability of laser propagation. Meanwhile, since the high energy laser heats up the air in the transmission path, the nonlinear thermal blooming effects further foment the optical effects from atmospheric turbulences [5]. Several researchers have been studying about the high energy laser propagation in the atmosphere for many years [6,7]. Moreover, single high energy beam cannot meet the tactical requirements during a fight; therefore, the spatial incoherent beam combining of multiple laser resources is used to enhance the intensity of laser on the target. This is because, the incoherently combined laser system has the advantages of simplicity, high efficiency, compactness, robustness, and no phase locking [8]. However, the significant achievements in this field so far focus on the effects of terrestrial atmospheric turbulence based on the Kolmogorov spectrum [9,10,11]. There is no adequate research on marine atmospheric turbulence. Recently, the research on pulsed laser technology has gained popularity. Relevant reports have demonstrated that the peak power of a nanosecond laser is capable of generating an output of 4.5 MW in 1 ns pulse [12,13]. Therefore, we have focused on the energy focusability of spatial incoherent beam combining of pulsed multi-laser propagation in the marine atmosphere.
The purpose of this study is to reveal the energy focusability of spatial incoherent beam combining of pulse laser in the marine atmosphere using numerical simulation. The rest of the paper is organized as follows. Section 2 presents the theoretical models of marine atmospheric turbulence and thermal blooming. Considering the complexity of marine atmospheric turbulence, the non-Kolmogorov power spectrum is adopted. Then, the thermal blooming of pulse laser is theoretically analyzed and the characterized expressions of energy focusability are described. Section 3 introduces the numerical simulation methods. We have considered the synthetical influence of atmospheric turbulence and thermal blooming. The power spectrum inversion method is used to simulate the atmospheric phase screens with the spectrum of different power indexes. Thermal blooming phase screens consider the mutual effects of multi-lasers on the air density. Section 4 presents the results of numerical simulations conducted on spatial incoherent beam combining for pulse laser propagation in the marine atmosphere. The simulated results of equivalent radius with long exposure, power in the bucket (PIB), Strehl ratio (SR), and peak value of intensity on the targets are presented. The principles of interaction between laser and marine atmosphere are explained. In particular, the scintillation index is used as a variable to study energy focusability. Section 5 discusses the effects of spectrum power index of non-Kolmogorov, pulse durations, and laser launch diameter on the energy focusability of spatial incoherent beam combining. Subsequently, to derive enough energy on the target, we have studied the equivalence between longer duration and shorter duration laser. Finally, the results of energy focusability of spatial incoherent beam combining for laser propagation in the marine atmosphere are summarized and future scope of research on the same is highlighted.
2. Theoretical models
2.1 Marine atmospheric turbulence
When laser propagates in the marine atmosphere, the atmospheric turbulence causes wave-front distortions of laser field. The modified power spectrum of atmospheric refractive index in the marine environment has been evaluated by several researchers [14,15,16]. The non-Kolmogorov expression is given by:
When $l_0 \rightarrow 0, L_0 \rightarrow \infty$, Eq. (6) represents the plane wave structure function based on Kolmogorov power spectrum of refractive index [21],
When $a_1=0, a_2=0$, Eq. (6) becomes the plane wave structure function based on von Karman power spectrum of refractive index given by:
Comparing Eq. (8) with [22], it can be seen that both are basically consistent in ignoring minor differences. When $D_{w-p}(\rho, L)=2$, Eq. (6) can be solved to obtain the atmospheric coherence length $r_0=2.1 \rho$.
Simulations of laser propagation in the atmosphere must describe the phase distortions of wave-front due to the atmospheric turbulence. The power spectrum method [23] is used to simulate the marine atmospheric phase screen. The phase power spectrum based on Eq. (1) is defined as:
The disadvantage of power spectrum method is that it has insufficient low frequency compensation. To rectify this defect, the subharmonic compensation is applied in the phase screen simulations. The index of the subharmonic compensation is selected according to the spectrum power index so that a best result is derived. Finally, the simulated statistic average values of two orders fit with the theoretical phase structure function.
2.2 Thermal blooming
Interactions of high energy laser with atmosphere results in heating up of air and change of air fluid state. The description of this situation involves the temperature, air density, and atmospheric pressure. The following differential equations represent the process of thermal blooming [24].
Effects of thermal blooming on laser propagation in the marine atmosphere include two aspects. First, thermal blooming intensifies the random fluctuations of atmospheric refractive index and laser beam spreading. Secondly, the random wind field causes a temperature gradient of atmosphere and weakens the thermal blooming effects. In this study, we have concentrated on the first aspect. At the constant pressure condition, Eq. (12) is simplified as: [25]
For a series of multiple pulses, when time of laser radiation is far less than the interpulse spacing, Eq. (13) is transformed to the following form [26]:
Equation (16) implies that wind plays an important role in the variation of air density. The wind across the laser beam causes the air density to shift along the wind direction. Before wind passes through the laser beam, air absorbs the energy of laser, and changes in air density are accumulated as the arrival of subsequent pulse. When laser span exceeds the transit time of wind across the laser beam, the subsequent thermal blooming attains steady state. The relation between refractive index and air density satisfies the Gladstone-Dale expression [24]. Furthermore, the wave-front fluctuation of light field in the laser transmission path is given by:
where $k_{G-D}$ is the Gladstone-Dale constant, $n_1$ denotes the fluctuation of air refractive index of approximately 1. According to Eq. (17), the phase screens of thermal blooming can be simulated. The thermal distortion parameter characterizes the strength of thermal blooming. The Bradley-Herrmann parameter [27] is used to describe the phase distortion of collimated beam of thermal blooming. For the focused and short pulse of Gauss intensity distribution, the thermal distortion parameter is expressed as [28]: where $G_s$ is the focus factor, $G_s=\left (\xi ^4-1-\ln \xi \right ) /\left [8(1-1 / \xi )^2\right ]$, $\xi =a_i / a_f$, $a_i$ is the beam initial characteristic radius, $a_i=D / 2 \sqrt {2}$, $D$ is the launch diameter of Gauss beam, $T_s$ is the thermal distortion parameter of short pulse beam given by [26]: where $n_T$ is the rate of refractive index as temperature, $n_T=-1.06 \mathrm {e}-06$ $\mathrm {K}^{-1}$ for the wavelength 1.064 μm under the standard condition, $\rho_0$is the air density, $c_p$ is the isobaric specific heat capacity, $L'$ is the distance of laser propagation, $E$ is the energy of single pulse, $E=P_0 \tau _p$, $P_0$ is the power of single pulse, $a_f$ is the spot radius on the focal plane without distortion, and $f\left (N_E\right )$ is the extinction factor given by: where $N_E=\alpha _e f$, $\alpha _e$ is the extinction coefficient, and $f$ is the laser focus. When $t=\tau _p$ and $L'=L$, the launch laser beam is close to Gauss shape and $T_{S F}$ can be regarded as the thermal distortion parameter.2.3 Expressions of energy focusability
When the high energy laser propagates in the marine atmosphere, the effects of atmospheric turbulence and thermal blooming occur simultaneously. This process can be described by paraxial equation [23] as follows:
Second is PIB. It indicates the energy percentage in a particular range of target and is given by:
where $R=a_f$. Third is SR, which is given by: where $I_p$ is the peak intensity after laser beam through atmosphere and $I_{0 p}$ denotes the peak intensity after laser beam through vacuum. Because the atmospheric turbulence is randomly fluctuating, $I_p$ varies with time.3. Numerical methods and parameters
3.1 Phase screens of atmospheric turbulence
Simulations of phase screen of atmospheric turbulence involve several factors. According to Eq. (4), the generalized $\tilde {C}_n^2$ is obtained. The statistical results of simulated phase screen must be consistent with Eq. (6). The power spectrum methods are based on Eq. (1) to Eq. (9). The subharmonic compensation with 3-25 orders is selected as the spectrum power index changes. The discrete expression of phase screen is:
The accuracy of phase screen simulation results has been verified by comparing with that of previous studies [27,28]. In this study, we have considered the following simulation parameters: grid number is 768$\times$768, interval distance is 2.5 mm, and the simulation step length is 100 m for the laser propagation distance of 10,000 m. We simulated 400 phase screens of atmospheric turbulence and verified the consistency between the simulated values and theoretical values $D(\rho )=D_{w-p}(\rho, L)$. Among them, two random samples are demonstrated in Fig. 1 ($\alpha =3.1, \tilde {\sigma }_R^2=0.0002$ at every 100 m) and Fig. 2 ($\alpha =3.9, \tilde {\sigma }_R^2=0.0002$ at every 100 m).
Fig. 1. Phase screens of atmospheric turbulence (left) and verification of simulated results (right) ($\alpha =3.1, \tilde {\sigma }_R^2=0.0002, p=3$).
Fig. 2. Phase screens of atmospheric turbulence (left) and verification of simulated results (right) ($\alpha =3.9, \tilde {\sigma }_R^2=0.0002, p=15$).
Figure 1 indicates that, when the spectrum power index is low, fitting of $D_{\varphi }(\rho )$ and $D(\rho )$ is not perfect due to deficiency of low-frequency compensation. In contrast, when the spectrum power index is high, this fitting is perfect as shown in Fig. 2. All the phase screens are examined to ensure that the overall situation is adequate to perform numerical simulations. In addition, since $\tilde {C}_n^2$ at the high power spectrum index is greater than that at the low power spectrum index, the tip-tilt of Fig. 2 appears bigger.
3.2 Phase screen of thermal blooming
Equation (17) indicates that the phase screens due to thermal blooming can be simulated. Some studies have proven that the parameters in simulations must meet the necessary condition in steady-state thermal blooming. For a focused Gauss beam, the grid interval must satisfy the following condition:
where $D$ is the laser launch diameter, $T_{S F}$ is the thermal distortion parameter in Eq. (18), $q_S$ is the converge factor, and $q_s=a_f / D$. It must be noted that $\Delta x$ varies with parameters, such as laser power, duration of single pulse, phase screen size, and laser launch diameter based on Eqs. (17)–(19) instead of fixed values. For the isobaric and steady state approximation, the distance between phase screens must satisfy the following expression, where $\sigma _n$ is the root-mean-square of refractive index fluctuation. In [29], considering the standard atmospheric conditions, the step length of laser propagation is considered as: where the value of $P_0$ is based on the above analysis in Eq. (19).Equations (30)–(32) indicate that, for a laser propagation distance of 10,000 m, the simulated parameters of thermal phase screens are set as: grid numbers are 1536$\times$1536 and 768$\times$768, grid interval is 1 mm and 2-2.5 mm for $\tau _p=0.5-1~\mu \mathrm {s}$ and $10-400 \mathrm {~ns}$, respectively, and step length of laser propagation is taken as 100 m. In the light of the above analysis, intensity distribution and thermal phase screen of the 20-th single laser pulse is simulated as shown in Fig. 3. These simulated parameters include $P_0=3 \times 10^7 \mathrm {~W}, \tau _p=1~\mu \mathrm {s}, L=10000 \mathrm {~m}, D=40 \mathrm {~cm}, \lambda =1.064~\mu \mathrm {m}, and \ v=5 \mathrm {~m} / \mathrm {s}$. The distortion parameter is calculated as $T_{S F}=93$ at $t=1~\mu \mathrm {s}$.
Fig. 3. Intensity distribution (right) at 10,000 m after 20 pulses through the atmosphere due to thermal blooming and thermal phase screen (left) of a single laser beam at 10,000 m.
Figure 3 indicates that thermal blooming induces the intensity distribution of laser, which resembles the shape of a kidney, and thermal phase screens drift along the direction of the wind. Owing to the equal time interval of laser pulses, the spacing between two kidney shaped distributions on the phase screen is equal.
3.3 Simulations of laser propagation
Solving Eq. (21), the solution is represented as:
We have represented several laser beams with circular pattern, which are symmetrically distributed and tangent to each other on the launch plane. The initial light fields are expressed as:
Fig. 4. Circular pattern of six laser beams on the launch plane.($R_1$=0.4 m, $\omega _0$=$W_0$, $W_0$=0.2 m)
It is well known that six combining beams are interactional and commonly cause air fluctuations when thermal blooming occurs. Figure 5 represents thermal blooming phase screen and intensity distribution of the 30-th laser pulse caused by the thermal blooming of six sub-beam combining at 10,000 m. The laser pattern in Fig. 4 and parameters in Table 1 are adopted.
Fig. 5. Intensity distribution (right) and thermal phase screen (left) of the six laser beams at 10,000 m after 30 pulses.
Table 1. Simulated parameters
Comparing Fig. 5 with Fig. 3, the reason for that six laser beams interplay, thermal phase screens and intensity distributions of Fig. 5 and Fig. 3 have a great difference. Therefore, the energy focusability of beam combining on the target has some severe changes. The distortion parameter is simply evaluated as $T_{S F}=558$ at $t=1~\mu \mathrm {s}$.
The following numerical simulations involve laser characteristics, marine atmospheric properties, and thermal blooming features. The simulated parameters are listed in Table 1.
In this study, we have focused on three aspects: First, thermal blooming of pulsed laser experiences from transient state to steady state. Second, air shifts along x-axis as wind blows, which moves the thermal phase screens by a certain grids M in the transit time of wind relative to single laser beam. Before and after steady state, the value of M is given by:
Considering the combination of six beams as shown in Fig. 4, the shift grids should be considered as (2 to 3)M. The number of pulses is $N_{p-w}=D / v$ before steady state of single laser beam. When steady state is attained, the density fluctuation of air is constant with time. Third, all combining laser beams with ideal beam quality factor are accurately and synchronously launched. The piston phase has been corrected during laser propagation. Additionally, the $R_L$, PIB, Ip and SR of laser spot on the target refer to the last pulse in the steady state of thermal blooming. Before this phase, these data are almost different. It should be noted that a large laser power is set here, which is significantly different from the actual situation [33]. However, this larger pulsed laser power can generate significant thermal blooming effects, which facilitates our theoretical understanding of the physical characteristics and behavior of them.
4. Results and analysis
4.1 Effect of atmospheric turbulence and thermal blooming on the energy focusability of pulse laser
From Table 1, laser focus $f=10,000 \mathrm {~m}$, scintillation index $\tilde {\sigma }_R^2=0.0011$ at every 100 m, spectrum power index of non-Kolmogorov $\alpha =3.6$, and distance of laser propagation $L$=10,000 m. For $D_{w-p}(\rho, L)=2$, solving Eq. (6), $r_0=76.65 \mathrm {~cm}$ and $D / r_0=0.5219$. Fig. 6 represents the intensity distributions caused by thermal blooming and atmospheric turbulence for $\tau _p=1~\mu \mathrm {s}, 200 \mathrm {~ns}$.
Fig. 6. Intensity distribution due to thermal blooming and atmospheric turbulence after 30 pulses. Thermal blooming at (a)$\tau _p=200 \mathrm {~ns}$ and (b)$\tau _p=1~\mu \mathrm {s}$. Both thermal blooming and atmospheric turbulence at (c) $\tau _p=200 \mathrm {~ns}$ and (d) $\tau _p=1~\mu \mathrm {s}$.
Figure 6 indicates that the thermal distortion of six-beam combining at $\tau _p=1~\mu \mathrm {s}$ is greater than that at $\tau _p=200 \mathrm {~ns}$. Simultaneously, atmospheric turbulence further intensifies spot spreading. The energy focusability of six-beam combining is shown in Table 2.
Table 2. Energy focusability of six-beam combining. a
The data in Table 2 indicate that the energy focusability of beam combining at $\tau _p=1~\mu s$ is easily affected by thermal blooming and atmospheric turbulence than that at $\tau _p=200 \mathrm {~ns}$. The equivalent radius with long exposure at $\tau _p=1~\mu s$ is greater than that at $\tau _p=200 \mathrm {~ns}$. The peak values of intensity distribution and SR at $\tau _p=1~\mu s$ drop several times than that at $\tau _p=200 \mathrm {~ns}$. PIBs at $\tau _p=1~\mu s$ are less than that at $\tau _p=200 \mathrm {~ns}$. In particular, the impact of atmospheric turbulence on the energy focusability is apparent. In the case of both thermal blooming and atmospheric turbulence, $R_L$ increases 3.8 times at $\tau _p=200 \mathrm {~ns}$ and 1.6 times at $\tau _p=1~\mu \mathrm {s}$. Simultaneously, PIB, $I_p$, and SR of energy focusability decrease in the range of 1 – 1.68 times than that of pure thermal blooming.
4.2 Effect of strength of atmospheric turbulence on energy focusability
To explore the effect of strength of atmospheric turbulence on energy focusability, the scintillation index is set as: $\tilde {\sigma }_R^2$=0.002, 0.0011, 0.0002 and non-Kolmogorov spectrum power index $\alpha =3.6$. We have used Eqs. (4) and (6) to calculate the generalized structure constant $\tilde {C}_n^2$=2.3249$\times 10^{-15}, 1.2787 \times 10^{-15}, 2.3249 \times 10^{-16} \mathrm {~m}^{3-a}$ and the atmospheric coherent length $r_0$=52.5 cm, 76.65 cm, 2.26 m, and $D / r_0$=0.76, 0.52, 0.177. Figure 7 shows that the strength of atmospheric turbulence affects the intensity distribution for the six-beam combining under the same thermal blooming.
Fig. 7. Intensity distribution due to thermal blooming and atmospheric turbulence after 30 pulses for $\tau _p=1 \mu \mathrm {s}$.
Figure 7 manifests that a stronger atmospheric turbulence causes a bigger spot spreading and weakens the peak values of intensity distribution. Table 3 lists the data of energy focusability of beam combining.
Table 3. Energy focusability of beam combining.
Table 3 indicates that the energy focusability of beam combining drops with an increase in atmospheric turbulence strength. When $\tilde {C}_n^2$ increases from $2.3249 \times 10^{-16} \mathrm {~m}^{3-a}$ to $2.3249 \times 10^{-15} \mathrm {~m}^{3-a}$, $R_L$ increases 1.31 times and $S R$ drops by 66 percent. Simultaneously, PIBs decrease 2.2 times and the peak value of intensity distribution drops by 66 percent. Therefore, strong atmospheric turbulence results in the decrease of energy focusability.
4.3 Effect of pattern of combining beams on energy focusability
To study the effect of pattern of combining beams on energy focusability, a rectangular pattern of six beams is represented in Fig. 8. When compared with Fig. 4, this pattern decreases the transit time of wind parallel to the $x$ axis.
Fig. 8. Rectangular pattern of six laser beams on the launch plane. $\left (\omega _0=\mathrm {w}_0, \mathrm {w}_0=0.2 \mathrm {~m}\right )$
In simulations, the basic parameters include laser focus $f=10,000 \mathrm {~m}$, scintillation index $\tilde {\sigma }_R^2=0.0011$ at every 100 m, spectrum power index of non-Kolmogorov $\alpha =3.6$, and distance of laser propagation $L$=10,000 m. Other parameters are listed in Table 1. Figure 9 simulates the intensity distributions of six-beam combining for the details in Fig. 8 due to thermal blooming and atmospheric turbulence for $\tau _p=1~\mu \mathrm {s}, 200 \mathrm {~ns}$ after 30 pulses.
Fig. 9. Intensity distribution of rectangular pattern due to thermal blooming and atmospheric turbulence for $\tau _p=1~\mu \mathrm {s}, 200 \mathrm {~ns}$.
Figure 9 indicates that the thermal distortion of rectangular pattern for six beam combining at $\tau _p=1~\mu \mathrm {s}$ is greater than that of the pulses at $\tau _p=200 \mathrm {~ns}$. Simultaneously, atmospheric turbulence further intensifies spot spreading. The energy focusability of beam combining is shown in Table 4.
Table 4. Energy focusability of laser beam combining. a
Compared to the pure thermal blooming, $R_L$ increases 4.5 times at $\tau _p=200 \mathrm {~ns}$ and 1.39 times at $\tau _p=1 ~\mu \mathrm {s}$ in the case of both thermal blooming and atmospheric turbulence. At the same time, PIB, $I_p$, and SR of the energy focusability decrease in the range of 1 – 1.92 times than that of pure thermal blooming as shown in Table 4. Comparing Table 4 with Table 2, the effects of rectangular and circular patterns on the energy focusability of beam combining are almost similar for the same parameters and there is no significant difference.
5. Discussions
5.1 Variation of the energy focusability as spectrum power index of non-Kolmogorov
To derive the variation of energy focusability as spectrum power index of non-Kolmogorov, the intensity distribution of pulse laser propagation are simulated from parameters listed in Table 1. First, the 12 points are calculated according to the theoretical models and an average of 10 numerical simulations for $\tau _p=1~\mu \mathrm {s}$ is performed based on the method discussed in Section 3. Then, the nonlinear least square method is used to fit these points. The four figures are shown as Fig. 10.
In Fig. 10(a), the equivalent radius with long exposure increases with spectrum power index of non-Kolmogorov in the case of $\tilde {\sigma }_R^2=0.002,0.0011,0.0002$. When $\alpha$ is greater than 3.7, $R_L$ fast goes up. When $\alpha$ is less than 3.7, $R_L$ increases slowly. In Fig. 10(b), (c), and (d), PIB, $I_p$, and SR decrease rapidly with an increase in $\alpha$. The general trends of change are the parabolic features. The scintillation index is lower, PIB, $I_p$, and SR are higher. The reasons for these are that $\tilde {C}_n^2$ becomes high with an increase in $\alpha$ for the atmospheric turbulence.
5.2 Impact of pulse duration on energy focusability
According to Eq. (16), the laser pulse duration is shorter and thermal blooming becomes weaker. However, the atmospheric turbulence is random, which leads to a decrease in the energy focusability of laser beam combining. We chose the pulse durations $\tau _p$ from 10 ns −1.5 $\mu \mathrm {s}$ to simulate impact of pulse duration on the energy focusability by the average values of 10 numerical simulations. Figure 11 represents the variation of energy focusability as laser pulse durations $\tau _p$.
In Fig. 11(a), the equivalent radius with long exposure increases with the laser pulse duration in the case of $\tilde {\sigma }_R^2$=0.002, and laser pulse duration is wider and $R_L$ is bigger. In Fig. 11(b), (c), and (d), PIB, $I_p$, and SR decrease rapidly with an increase in laser pulse durations and are characterized by exponential change. Because the spectrum power index of non-Kolmogorov $\alpha$ relates to $\tilde {C}_n^2$, when $\alpha$ is high, atmospheric turbulence becomes strong. This weakens PIB, $I_p$, and SR.
5.3 Effect of laser launch diameter on energy focusability
Laser launch diameter is the characteristic quantity that is important for laser beam combining. We considered laser launch diameters D in the range of 20 – 48 cm to simulate the effects of laser launch diameter on energy focusability based on the above simulation methods. During the processing, the circular pattern of six laser beams are considered. Figure 12 shows the variation of energy focusability with laser launch diameters.
In Fig. 12(a), the equivalent radius with long exposure increases as laser launch diameters in the case of $\tilde {\sigma }_R^2$=0.002, and laser launch diameter is higher and $R_L$ is smaller. But when D is greater than 32 cm, this change becomes flat. In Fig. 12(b), (c), and (d), PIB, $I_p$, and SR increase with an increase in laser launch diameters. That is, the big laser launch diameters are beneficial to the energy focusability of spatial incoherent beam combining. This is because, the spectrum power index of non-Kolmogorov $\alpha$ relates to $\tilde {C}_n^2$, the higher values of laser launch diameters weaken the effect of atmospheric turbulence. Therefore, increasing laser launch diameter is helpful to enhance the energy focusability of laser beam combining. Increment of energy focusability is high when $\alpha$ is low.
5.4 Considerations of pulse laser repetition
The above analysis reveals that the shorter pulse duration and bigger laser launch diameter are beneficial to enhance the energy focusability of spatial incoherent beam combining in thermal blooming and weak atmospheric turbulence. In contrast, the higher spectrum power index of non-Kolmogorov causes a stronger atmospheric turbulence, which goes against energy focusability. However, the short pulse duration is adverse enough to provide sufficient energy on the target, and laser launch diameter is restricted by conditions. To derive sufficeint energy on the target, the short pulse laser is necessary to increase repetition. We simulated the energy focusability of spatial incoherent beam combining in the light of above methods for $\tau _p=100 \mathrm {~ns}, 1~\mu \mathrm {s}$ at $\tilde {\sigma }_R^2$=0.002. The pulse laser repetition of $1~\mu s$ and $100 \mathrm {~ns}$ is respectively Rp=125 Hz and Rp=1250 Hz. The non-Kolmogorov spectrum power indexes are considered as $\alpha =3.3,3.6,3.9$. Figure 13 describes the variations of energy focusability as the number of pulses. In this study, the radius of the ring energy is considered as R=5$a_f$.
Figure 13 shows that the energy focusability of spatial incoherent beam combining can obtain steady states with an increase in the number of pulses. When the pulse laser repetition of $1~\mu s$ and $100 \mathrm {~ns}$ is matching, the same energy can be derived on the target; however, the PIB and SR of spatial incoherent beam combining have no a great distinction at a steady state. Results indicate that the pulse laser of $1~\mu s$ duration attain steady state at 27 pulses, the pulse laser of $100 \mathrm {~ns}$ duration attains steady state at 270 pulses. After steady states, this difference is approximately a few percentages in PIB and less than one percentage in SR as shown in Fig. 13. In the spatial incoherent beam combining, SR is more easily affected by atmospheric turbulence than PIB and becomes fluctuant before steady states. However, the overall change is not significant. According to the above analysis, we have every reason to infer that increasing the repetition of shorter pulse is almost equivalent to the effect of longer pulse influenced by atmospheric turbulence and thermal blooming in energy focusability.
6. Conclusions
This study discussed the energy focusability of spatial incoherent beam combining for pulse laser propagation in the marine atmosphere by numerical simulations. We introduced the power spectrum of atmospheric refractive index in the marine environment with generalized spectrum power index $\alpha$, which operated in the range of $3.05 \leq \alpha \leq 3.95$. Next, we studied the focused six pulse laser beam influenced by thermal blooming and atmospheric turbulence. We stimulated the marine atmospheric phase screen and thermal blooming phase screen and investigated the effects of spectrum power indexes of non-Kolmogorov, pulse durations, laser launch diameters, and patterns of combining beams on the energy focusability of spatial incoherent beam combining. Further, the effect of pulse laser repetition on laser energy on the target is discussed. In particular, we considered the interactions of six sub-beam combining in thermal blooming. Results show that the interactions of six sub-beam combining result in the spots of thermal blooming, which are different from pure focused cases. In addition, the effects of rectangular and circular patterns on the energy focusability of beam combining indicated no significant difference under the same parameters. The details presented in this study have been summarized as follows. First, PIB, $I_p$, and SR of pulse beam combining decreased with an increase in $\alpha$; however, $R_L$ operated in reverse mode. Theoretically, $\tilde {C}_n^2$ increased with an increase in $\alpha$ for same thermal blooming. Second, the small pulse durations indicated a drop in thermal blooming and enhanced the energy focusability of spatial incoherent beam combining under the same atmospheric turbulence. However, laser of small duration is not helpful to derive the high ring energy on the target. Third, the proper laser launch diameter was beneficial to the energy focusability of spatial incoherent beam combining. Our simulations in Fig. 12 proved this aspect. In fact, increasing the laser launch diameter is favorable but not significant. In addition, increasing the repetition of pulse laser for shorter durations could not effectively improve the energy focusability of spatial incoherent beam combining. Numerical simulations showed that the PIB and SR of spatial incoherent beam combining had no distinction at a steady state between laser with repetition at 125 Hz and 1250 Hz. Therefore, to derive the same energy on the target, the $\mu s$ order pulse laser (or higher duration pulse laser) and ns order pulse laser were almost equivalent. However, increasing the laser launch diameter appropriately would be helpful.
Appendix A
The scintillation index of plane wave of laser propagating in the marine atmosphere.
According to $\cos (x)=(R e) \exp (-i x)$, taking $Q_H=L \kappa _m^2 / k$, the following expression is deduced.
Assuming $a_0=1, a_1, a_2$, corresponding to $\gamma =0,1,3-\alpha / 2$, a unitive expression is represented as:
Utilizing $U(a ; c ; z)=\frac {1}{\Gamma (a)} \int _0^{\infty } \exp (-z t) t^{a-1}(1+t)^{c-a-1} d t$, Eq. (39) can be written as:
Because of $U(a ; c ; z)=\frac {\Gamma (1-c)}{\Gamma (1+a-c)}{ }_1 F_1(a ; c ; z)+\frac {\Gamma (c-1)}{\Gamma (a)} z^{1-c}{ }_1 F_1(1+a-c ; 2-c ; z)$, this yields
From ${ }_1 F_1(a ; b ; z) \simeq 1+\frac {a}{b} z(z \ll 1)$, the following approximation is obtained.
Similarly, the following can be derived.
Comprehensively, considering Eq. (42) and Eq. (43), the following expression is written as:
When $\alpha =11 / 3$, substituting Eq. (44) into Eq. (38), which results in
Appendix B
The plane wave structure function of laser propagating in the marine atmospheric turbulence. The wave structure function of plane wave is defined as:
Equation (46) can be regarded as the sum of three terms from (•) $\left [1+a_1 \frac {\kappa }{\kappa _m}+a_2\left (\frac {\kappa }{\kappa _m}\right )^{3-\alpha / 2}\right ]$. Thus, Eq. (46) can be rewritten as:
Comparing Eq. (47) with Eq. (46), the corresponding terms are respectively given by:
Therefore, a unitive expression is obtained for the above three terms,
According to the confluent hypergeometric function of second kind, the following expression is given.
where $\Gamma (\bullet )$ denotes gamma function. Using Eq. (53), Eq. (52) is transformed to:With the following relation expressions,
$\Gamma (a+1-n)=\frac {(-1)^n \Gamma (a+1)}{(-a)_n}$, $\Gamma (n+a)=(a)_n \Gamma (a),(a-n)_n=(-1)^n(1-a)_n$, $(a)_{2 n}=(a)_n(n+a)_n$ and $(a)_{2 n}=2^{2 n}\left (\frac {1}{2} a\right )_n\left (\frac {1}{2}+\frac {1}{2} a\right )_n$. Eq. (54) can be represented as:
From the following approximations, ${ }_3 F_3(a, b, c ; d, e, f ; z) \simeq 1+\frac {a b c}{d e f} z \approx 1(z \ll 1),{ }_1 F_1(a ; b ; z) \simeq 1+\frac {a}{b} z(z \ll 1),$ ${ }_1 F_2\left (1+\frac {\gamma }{2} ; 2-\frac {\alpha }{2}+\frac {\gamma }{2}, 1 ; \frac {\kappa _0^2 \rho ^2}{4}\right ) \simeq 1+\frac {1+\frac {\gamma }{2}}{2+\frac {\gamma }{2}-\frac {\alpha }{2}} \frac {\kappa _0^2 \rho ^2}{4} \approx 1+\frac {\kappa _0^2 \rho ^2}{4}\left (\frac {\kappa _0^2 \rho ^2}{4} \ll 1\right )$.
Eq. (55) is reduced to
For $\frac {\kappa _m^2 \rho ^2}{4} \gg 1,{ }_1 F_1\left (1+\frac {\gamma }{2}-\frac {\alpha }{2} ; 1+\frac {\gamma }{2} ;-\frac {\kappa _m^2 \rho ^2}{4}\right ) \simeq \frac {\Gamma \left (1+\frac {\gamma }{2}\right )}{\Gamma \left (\frac {\alpha }{2}\right )}\left (\frac {\kappa _m^2 \rho ^2}{4}\right )^{\frac {\alpha }{2}-\frac {\gamma }{2}-1}$. Eq. (56) is rewritten as:
In the cases of $\gamma =0,1,3-\alpha / 2$ and $a_0=1, a_1, a_2$, the plane wave structure function is represented by:
Equation (58) represents the plane wave structure function including inner scale and outer scale. This equation is useful to comprehensively know the change laws affected by the marine atmospheric turbulence. From Eq. (58), the plane wave structure function is influenced by inner scale, outer scale, and spectrum power index of non-Kolmogorov.
Funding
State Key Laboratory of Pulsed Power Laser Technology (SKL2020KF06); Key projects of science foundations of Anhui Education Department (KJ2019A0619); Natural Science Foundation of Anhui Province (2208085QD104).
Acknowledgments
I would like to express my gratitude to Dr. Ruizhong Rao and Dr. Jianzhu Zhang who discussed this work with me during the writing of this paper.
Disclosures
The authors declare no conflicts of interest.
Data availability
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.
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