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High-precision calculation and experiments on the thermal blooming of high-energy lasers

Qi Zhang, Qili Hu, Hongyan Wang, Ming Hu, Xingyu Xu, Jingjing Wu, and Lifa Hu

Open Access Open Access

Abstract

Thermal blooming (TB) is one of the important factors affecting the quality of high-energy laser beams. Reasonable simulation of thermal blooming is important to the application of a high-energy laser. However, reported investigations on TB simulation are mainly based on one method, such as the perturbation method or the phase screen method, which often leads to obvious errors in some conditions. In the paper, the reasonable ranges of optical generalized distortion parameters for both methods are determined based on the reported experimental data, which solves the problem of accurate TB simulations for the first time. In addition, the dynamic effect of thermal blooming is also calculated. Finally, the formula method is presented to extract the phase of thermal blooming distortion. We then use LC-SLM (Liquid crystal spatial light modulator) to emulate thermal blooming effect in the lab. The experimental results are more consistent with the numerical simulation results than conventional phase extraction methods. Our work provides a quantitatively and programmable way to accurately simulate TB with LC-SLM in the lab.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

The thermal blooming caused by the interaction with air severely limited the efficiency of high energy laser (HEL) transmission in the atmosphere [1–5]. The thermal blooming effect will cause the energy dissipation of the laser beam and reduce the beam quality, which limits the application and development of HEL [6–20]. Therefore, it is of great significance to study the thermal blooming effect of HEL propagation in the atmosphere [21–24].

The field experiments to study the transmission of HEL in the atmosphere require many persons and expensive equipment, such as high-energy laser, IR CCD and so on. In addition, the atmospheric turbulence conditions are not controllable. Therefore, the numerical simulation and laboratory emulations based on the transmission characteristics of HEL in the atmosphere have important value of research. Many researches have been reported on the thermal blooming effect in laser transmission for decades [25–36]. In 2005, M. H. Mahdieh et al. simulated ring pattern HEL beam when propagating through a high-energy CO2 gas. The simulation was performed using “ray optics” theory as well as “wave optics”. It was concluded that the ray optics cannot predict the simulation precisely for the conditions of lower Fresnel numbers [37]. In 2013, Xiaoling Ji et al. studied the changes of the intensity distribution, the centroid position and the mean-squared beam width of an Airy beam propagating through the atmosphere. It was shown that the thermal blooming resulted in a central dip of the center lobe, and caused the center lobe to spread and decrease. In contrast with the center lobe, the side lobes were less affected by thermal blooming, such that the intensity maximum of the side lobe might be larger than that of the center lobe [38]. In 2016, S. A. Shlenov et al. studied the influence of wind speed on the focusing ability of CO2 lasers radiation during transmission in the atmosphere. It showed that wind speed could reduce the influence of the thermal blooming effect, and then improve the focusing ability of laser beam [39]. In 2020, Mark F. Spencer studied the influence of thermal blooming and turbulent effects on logarithmic amplitude variance and branch point density based on Gaussian beams [40,41]. In 2021, Lu Zhao et al. used the phase screen method to study the influence of the propagation distance, initial power, wind speed, atmospheric absorption coefficient, initial radius and wavelength of the beam on the steady-state thermal blooming effect of the vortex beam in atmospheric transmission [42]. In 2022, Xiangyizheng Wu et al. studied the thermal blooming effect of high-energy lasers transmitted in air, helium and nitrogen, respectively. Their results indicate that nitrogen can suppress the thermal blooming effect [43].

In these reported works, popular simulation methods for thermal blooming can be classified as follows: perturbation method (PM) and phase screen method (PSM). Usually one of these methods was adopted to estimate the thermal blooming effect under various conditions without considering its limitations. This paper will compare the two methods based on the experimental results reported in the literature, and determine their reasonable application conditions. At the same time, accurate phase extraction method is presented, and the programmable liquid crystal spatial light modulator LC-SLM as the thermal blooming LC simulator is used to realize the laboratory emulation of thermal blooming. It has the advantage of being quantitative and repeatable.

In section 2, the numerical simulations of steady-state thermal blooming are given including the two different theoretical models and their results of different influencing factors. The results are compared and analyzed with the experimental results reported in the literature [33], so as to determine the effective simulation interval of different simulation methods. In section 3, the numerical simulation of dynamic thermal blooming is completed by analyzing the time-related variables among the influencing factors of thermal blooming. In section 4, the liquid crystal thermal blooming simulation system is designed in the laboratory, and results are discussed. Finally, the conclusions are given in section 5.

2. Numerical simulation of steady-state thermal blooming

2.1 Theoretical model of steady-state thermal blooming

According to PM, the distortion caused by the thermal blooming effect is considered as a simple disturbance item on the original transmission beam. It is a geometric optics method and ignores the influence of the diffraction effect. If the initial laser beam is a Gaussian beam, the first-order perturbation solution of the laser transmission steady-state thermal blooming can be expressed as

$$\begin{aligned} \frac{{I({x,y,z} )}}{{{I_u}({x,y,z} ){e^{ - \alpha z}}}} &= \textrm{exp} \left\{ { - {N_c}\left[ {2\left( {\frac{x}{a}} \right){e^{[{ - ({{x^2} + {y^2}} )/{a^2}} ]}} + \frac{{\sqrt \pi }}{2}{e^{({ - {y^2}/{a^2}} )}}\left( {1 - 4\frac{{{y^2}}}{{{a^2}}}} \right)\left( {1 + erf\left( {\frac{x}{a}} \right)} \right)} \right]} \right\}\\ {N_c} &= \left( {\frac{{ - 2dn/dT{I_0}z}}{{{n_0}\rho {C_p}va}}} \right)\left[ {1 - \frac{{({1 - {e^{ - \alpha z}}} )}}{{\alpha z}}} \right] \approx \left( {\frac{{ - dn/dT{I_0}{z^2}\alpha }}{{{n_0}\rho {C_p}va}}} \right),\;\; \alpha z \ll 1 \end{aligned}$$
where NC is the thermal blooming distortion parameter, dn/dT is the change rate of the refractive index, I0 is the initial laser intensity, z is the transmission distance, n0 is the atmospheric refractive index, ρ is the air density, Cp is the specific heat capacity of the gas at constant pressure, v is the wind speed, a is the beam radius, and α is the medium absorption coefficient.

In PSM, there are equivalently multiple phase screens between the high power laser source and the image. The Fourier transform method is used between two adjacent screens to obtain the light field distribution on the second screen, which leads to the final light field distribution. The Fourier transform method for solving between two adjacent screens can be understood as: the wave propagates for half step, and then the medium propagates for the next half step, and the vacuum wave equation is solved to obtain the field on one step. The paraxial beam scalar wave equation can be expressed as:

$$2ik\frac{{\partial \varphi }}{{\partial z}} = \nabla _T^2\varphi + {k^2}\left[ {\frac{{{n^2}({x,y,z,t} )}}{{n_0^2}} - 1} \right]\varphi .$$

Let φn be the field value at zn, the solution φn + 1 at zn + 1 = zn+Δz can be expressed as:

$${\varphi ^{n + 1}} = {\varphi ^n}\textrm{exp} \left[ {\frac{i}{{2k}}({\nabla_T^2 + {k^2}\delta \bar{\varepsilon }} )\Delta z} \right]$$
where $\delta \bar{\varepsilon } = \frac{1}{{\Delta z}}\int_{{z^n}}^{{z^n} + \Delta z} {\delta \varepsilon ({z^{\prime}} )dz^{\prime}} $, $\delta \varepsilon = 0.46{\rho _1}$, ${\rho _1} = \frac{{({\gamma - 1} )\alpha }}{{c_s^2v}}\int_{ - \infty }^x {I({x^{\prime},y,z} )dx^{\prime}} $. Make $\varphi ({x,y,z} )$ discrete Fourier transform:
$$\varphi ({x,y,z} )= \sum\limits_{m ={-} \frac{N}{2}}^{\frac{N}{2} - 1} {\sum\limits_{n ={-} \frac{N}{2}}^{\frac{N}{2} - 1} {{\varphi _{mn}}(z )\textrm{exp} \left[ {\frac{{i2\pi ({mx + ny} )}}{L}} \right]} } .$$

Perform Fourier transform on the light field to get the light wave expression:

$${\varphi _{mn}}({z,t} )= \frac{1}{{{N^2}}}\sum\limits_{j = 0}^{N - 1} {\sum\limits_{i = 0}^{N - 1} {\varphi ({j\Delta x,i\Delta y,z} )\textrm{exp} \left[ {\frac{{ - i2\pi ({mj + ni} )}}{N}} \right]} } .$$

The half-step length of light wave propagation is expressed as:

$${\varphi _{mn}}\left( {z + \frac{{\Delta z}}{2},t} \right) = {\varphi _{mn}}({z,t} )\textrm{exp} \left[ { - \frac{i}{{2k}}{{\left( {\frac{{2\pi }}{L}} \right)}^2}({{m^2} + {n^2}} )\frac{{\Delta z}}{2}} \right].$$

The expression of converting light wave into light field is:

$$\varphi \left( {j\Delta x,i\Delta y,z + \frac{{\Delta z}}{2},t} \right) = \sum\limits_{m ={-} \frac{N}{2}}^{\frac{N}{2} - 1} {\sum\limits_{n ={-} \frac{N}{2}}^{\frac{N}{2} - 1} {{\varphi _{mn}}\left( {z + \frac{{\Delta z}}{2},t} \right)\textrm{exp} \left[ {\frac{{ - i2\pi ({mj + ni} )}}{N}} \right]} } .$$

Considering the influence of the medium, the light field is expressed as:

$$\varphi ^{\prime}\left( {j\Delta x,i\Delta y,z + \frac{{\Delta z}}{2},t} \right) = \varphi \left( {j\Delta x,i\Delta y,z + \frac{{\Delta z}}{2},t} \right)\textrm{exp} \left( {\frac{{ik}}{2}\delta \bar{\varepsilon }\Delta z} \right).$$

The expression for converting light field to light wave is:

$$\varphi {^{\prime}_{mn}}({z,t} )= \frac{1}{{{N^2}}}\sum\limits_{j = 0}^{N - 1} {\sum\limits_{i = 0}^{N - 1} {\varphi ^{\prime}\left( {j\Delta x,i\Delta y,z + \frac{{\Delta z}}{2},t} \right)\textrm{exp} \left[ {\frac{{ - i2\pi ({mj + ni} )}}{N}} \right]} } .$$

The half-step length of light wave propagation is expressed as:

$$\varphi {^{\prime}_{mn}}({z + \Delta z,t} )= \varphi {^{\prime}_{mn}}({z,t} )\textrm{exp} \left[ { - \frac{i}{{2k}}{{\left( {\frac{{2\pi }}{L}} \right)}^2}({{m^2} + {n^2}} )\frac{{\Delta z}}{2}} \right].$$

The light field distribution of one step length is obtained as:

$$\varphi ^{\prime}({j\Delta x,i\Delta y,z + \Delta z,t} )= \sum\limits_{m ={-} \frac{N}{2}}^{\frac{N}{2} - 1} {\sum\limits_{n ={-} \frac{N}{2}}^{\frac{N}{2} - 1} {\varphi {^{\prime}_{mn}}({z + \Delta z,t} )\textrm{exp} \left[ {\frac{{ - i2\pi ({mj + ni} )}}{N}} \right]} } .$$

The entire transmission process can be obtained by analogy from the solution process of the above step.

2.2 Results of steady-state thermal blooming

When there is a transverse wind, it can be regarded as the thermal blooming effect of laser transmission under forced convection. When the air passes through the laser beam along the wind direction, the molecules in the air continuously absorb the laser energy. The temperature of this part of the air becomes higher and higher, resulting in a decrease in the refractive index. So far, the commonly used methods for simulating the thermal blooming effect are PM and PSM. The following will compare the different simulation results of these two methods under the same conditions. The initial parameter is set to: λ = 10.6µm, P = 50 kW, a = 0.2 m, α = 6.5 × 10−5 /m, v = 5 m/s, z = 5 km. And the emitted laser beam is a Gaussian beam. The influence of changing various parameters on distortion under different simulation methods will be discussed below.

First, the transmission distances as the variable are 1 km, 3 km, 5 km, 10 km, respectively. The corresponding N values are 0.8559, 2.3897, 3.6762, 6.3501, respectively. With PM and PSM, the normalized light intensity (defined as the ratio of received light intensity/received light intensity maximum) contours are as shown in Fig. 1. With the increase of transmission distance, the distortion parameter Nc increases. And the beam gradually becomes crescent shape. Under the same conditions, the simulation results of both methods are different. And PSM shows relatively strong distortion effect when the distance is less than 5 km.

 figure: Fig. 1.

Fig. 1. Contour diagrams of light intensity simulated by the two methods at different transmission distances. (a1), (b1), (c1) and (d1) are the results of PM. (a2), (b2), (c2) and (d2) are the results of PSM. The transmission distance increases from 1 km to 10 km from first column to the 4th one, respectively.

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Next, the initial laser powers as the variable are 10 kW, 40 kW, 60 kW, 100 kW, respectively. The corresponding N values are 0.7352, 2.9410, 4.4114, 7.3524, respectively. With PM and PSM, the normalized light intensity contours are as shown in Fig. 2. With the increase of the initial laser power, the energy absorbed by the atmosphere in the transmission process, the temperature of the medium and the change of refractive index increases. The beam gradually becomes crescent shape. Under the same conditions, the simulation results of both methods are also different. The simulation results of PSM show relatively strong distortion effect when the generalized distortion parameters are small.

 figure: Fig. 2.

Fig. 2. Contour diagrams of light intensity under different initial laser powers simulated by the two methods. (a1), (b1), (c1) and (d1) are the results of PM. (a2), (b2), (c2) and (d2) are the results of PSM. The initial laser power increases from 10 kW to 100 kW from the first column to the 4th one, respectively.

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Thirdly, the wind speed as the variable is 2 m/s, 5 m/s, 7 m/s, 10 m/s, respectively. The corresponding N values are 9.1859, 3.6762, 2.6261, 1.8384, respectively. With PM and PSM, the normalized light intensity contours are as shown in Fig. 3. As the wind speed increases, the distortion parameter Nc decreases. The degree of distortion is weakened, and the image gradually changes from the initial crescent shape to a circle. Under the same conditions, the simulation results of different methods are different. For v is larger than 5 m/s, the simulation results of PM have no obvious beam expansion phenomenon, but the simulation results of PSM show relatively strong distortion effect.

 figure: Fig. 3.

Fig. 3. Contour maps of light intensity at different wind speeds simulated by the two methods. (a1), (b1), (c1) and (d1) are the simulation results of the perturbation method. (a2), (b2), (c2) and (d2) are the simulation results of PSM. The wind speed is ranged from 2 m/s to 10 m/s from the first column to the 4th one, respectively.

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In short, the thermal blooming effect will increase with the increase of the transmission distance and the initial laser power, thereby reducing the laser transmission efficiency. While the thermal blooming effect will be weakened with the increase of wind speed, thus improving the laser transmission efficiency. On the other hand, by comparing the above numerical simulation results, different numerical simulation methods have obvious differences under the same simulation parameters. Therefore, it is necessary to clarify quantitatively the reasonable application conditions for different numerical simulation methods.

2.3 Reasonable application conditions for PM and PSM

Although PM is simple, it will not accurate when the distortion parameter is larger than a certain value because it is mainly based on ray optics. At the same time, PM can not interpret the phenomenon of spot diffusion. Phase screen method is widely used nowadays, but its distortion effect is too obvious when the distortion value is very small. Therefore, this paper combines these two methods to find the most suitable range of application for each method, which will improve the accuracy of numerical simulation. The experimental results were reported in the article Twenty-Five Years of Thermal Blooming: An Overview in 1990 [33], as shown in Fig. 4. IREL refers to the relative peak intensity (maximum received light intensity/maximum initial light intensity), and N refers to the generalized distortion parameter. The experimental results are in good agreement with the empirical model, and its error is around 0.3. Therefore, we could use the empirical model to determine the applicable range of each method. It could be calculated as follows:

$${I_{REL}} = \frac{1}{{1 + 0.0625{N^2}}}.$$

In order to compare with the empirical model, the abscissa is the generalized distortion parameter N. Below will draw pictures about relative peak intensity IREL depending on N under different numerical simulation methods. The expression of the generalized distortion parameter is

$$\begin{array}{c} N = {N_c}f({{N_E}} )q({{N_F}} )s({{N_\omega }} )\\ q({{N_F}} )= \left[ {\frac{{2{N_F}^2}}{{{N_F} - 1}}} \right]\left[ {1 - \frac{{In{N_F}}}{{{N_F} - 1}}} \right]\\ f({{N_E}} )= \frac{2}{{{N_E}^2}}[{{N_E} - 1 + \textrm{exp} ({ - {N_E}} )} ]\\ s({{N_\omega }} )= \frac{2}{{{N_\omega }^2}}[{({{N_\omega } + 1} )In({{N_\omega } + 1} )- {N_\omega }} ]\end{array}$$
where f(NE) is the extinction number correction factor, q(NF) is the Fresnel number correction factor, s(Nω) is the beam scanning correction factor, and Nc is the steady-state thermal blooming distortion parameter of the collimated beam. They can be expressed as:
$$\begin{array}{c} {N_c} = \frac{{16\sqrt 2 ({ - {n_T}} )\alpha P{L^2}}}{{\pi {n_0}\rho {c_p}v{D^3}}}\\ {N_E} = ({\alpha + s} )z\\ {N_F} = {{k{a^2}} / z}\\ {N_\omega } = \frac{{\omega z}}{{{v_0}}} \end{array}.$$

 figure: Fig. 4.

Fig. 4. Blooming IREL dependence on the parameter N for experimental and simulated results. (Reference [33], Fig. 6)

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In order to evaluate the reasonable range of each method more intuitively, we introduce the reference of error, which can be expressed as:

$$\begin{aligned} {I_{error1}} &= {I_{PM}} - {I_{EM}}\\ {I_{error2}} &= {I_{PSM}} - {I_{EM}} \end{aligned}.$$

Among them, Ierror represents the difference between simulation results and experimental results, IPM represents the normalized peak intensity simulated by PM, IPSM represents the normalized peak intensity simulated by PSM, and IEM represents the normalized peak intensity of empirical model.

The transmission distance is ranged from 1 km to 50 km, and the step size is 0.1 km during simulation. Figure 5(a) shows the relationship between the relative peak intensity and the generalized distortion parameter generated by PM and PSM under different transmission distances, and compares them with empirical model. Figure 5(b) shows the error analysis between the relative peak intensity generated by the two numerical simulation methods at different transmission distances and the reference line. It can be seen from Fig. 5 that for N < 4.8, the error of PM is small. And for N > 4.8, the error of PSM is small.

 figure: Fig. 5.

Fig. 5. (a) normalized peak intensity as a function of N. perturbation: PM; phase screen: PSM. (b) the errors of normalized peak intensity as a function of N at different transmission distances for PM and PSM.

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The initial laser power is ranged from 1 kW to 300 kw, and the step size is 0.5 kW. Figure 6(a) shows the relative peak intensity IREL as a function of the generalized distortion parameter generated by the two methods for different initial laser powers. Figure 6(b) shows the error analysis between the normalized peak intensity, IREL generated by the two numerical simulation methods at different initial laser powers and the reference line. It can be seen from Fig. 6 that for N < 4.8, the error of PM is small. And for N > 4.8, the error of PSM is small.

 figure: Fig. 6.

Fig. 6. (a) Normalized peak intensity as a function of N for PM and PSM at different initial laser powers. perturbation: PM; phase screen: PSM. (b) the errors of normalized peak intensity as a function of N at different initial laser powers.

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The wind speed is ranged from 0.5 m/s to 30 m/s, and the step size is 0.2 m/s. Figure 7(a) shows the relationship between the relative peak intensity and the generalized distortion parameter generated by the two numerical simulation methods under different wind speeds, and compares them with empirical model. Figure 7(b) shows the error analysis between the normalized peak intensity (maximum received light intensity/maximum initial light intensity) generated by the two numerical simulation methods at different wind speeds and the reference line. It can be seen from Fig. 7 that for distortion parameter N < 4.8, the error of PM is smaller. And for distortion parameter N > 4.8, the error of PSM is smaller.

 figure: Fig. 7.

Fig. 7. (a)Normalized peak intensity as a function of N for PM and PSM at different wind speeds. perturbation: PM; phase screen: PSM. (b) is the error of normalized peak intensity as a function of N at different wind speeds.

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Although the intervals selected for the distortion parameter N corresponding to each influencing factor are different, their values are all relatively close. The error of two models is small near the parameter N corresponding to the intersection of the simulation curves of different models. The following conclusions are drawn: For distortion parameter N < 4.8, the error of PM is smaller; And for distortion parameter N > 4.8, the error of PSM is smaller. Therefore, distortion parameter N of 4.8 could be determined as the threshold, which makes the error between the numerical simulation results and the reference line comparison less than 0.2. The feasibility of this method could be validated. After analysis, the source of the simulation error may include the following aspects: (1) The atmospheric absorption coefficient α was set as a fixed value in the simulation process, so the atmospheric absorption coefficient change caused by the atmospheric temperature change in the laser transmission process was ignored. (2) The experimental results are irregularly distributed near the empirical model as shown in Fig. 4, so there is a certain error for the empirical model. (3) The experimental environment of laser transmission in the atmosphere is too complex, and there are some uncontrollable factors and nonlinear phenomenon. So the errors less than 0.2 are within the acceptable range.

3. Dynamic thermal blooming simulation

Although dynamic thermal blooming simulation is rarely reported, it is important to the application of HELs. In fact, the absorption coefficient is time dependent. There are two reasons for the change of absorption coefficient. On the one hand, the laser beam collides with gas molecules during transmission. The gas molecules absorb heat rapidly, which makes the air temperature rise. And then the absorption coefficient changes. On the other hand, aerosol particles absorb heat during laser transmission, and then release heat into the air, which makes the air temperature rise and the absorption coefficient change. The change in this part of the absorption coefficient is called the aerosol effective absorption coefficient.

$$\begin{array}{c} {\alpha _{eff}}(t )= \int {{Q_{abs}}(a )\pi {a^2}n(a )\left\{ {1 - \textrm{exp} \left( {\frac{{ - t}}{{\tau (a )}}} \right)} \right\}} da\\ n(a )= \left\{ \begin{array}{l} 2 \times {10^{ - 12}}{a^{ - 4}},2 \times {10^{ - 2}} < a < 30\mu m\\ 0,elsewhere \end{array} \right.\\ \tau (a )= \frac{{{a^2}{\rho _p}{c_p}}}{{3{k_a}}} \end{array}.$$
Where Qabs(a) is the absorption efficiency factor, n(a) is the number of aerosol particles with a radius of a per unit volume, τ(a) is the time for the particles to absorb heat and reach the final temperature, ρp is the particle density, and cp is the specific heat of the particles, ka is the thermal conductivity of air.
$$\begin{array}{c} {Q_{abs}} = \frac{{{\sigma _{abs}}}}{{\pi {r^2}}} = {Q_{ext}} + {Q_s}\\ {Q_{ext}} = \frac{2}{{{x^2}}}\sum\limits_{n = 1}^\infty {({2n + 1} ){\rm{Re}} ({{a_n} + {b_n}} )} \\ {Q_s} = \frac{2}{{{x^2}}}\sum\limits_{n = 1}^\infty {({2n + 1} )({{{|{{a_n}} |}^2} + {{|{{b_n}} |}^2}} )} \end{array}$$
where Qext is the extinction efficiency factor, Qs is the scattering efficiency factor, σabs is the absorption cross section, an and bn are the meter scattering coefficients, which can be expressed as
$$\begin{aligned} {a_n} &= \frac{{{\Psi _n}(x ){\Psi _n}^{\prime}({mx} )- m{\Psi _n}^{\prime}(x ){\Psi _n}({mx} )}}{{{\zeta _n}(x ){\Psi _n}^{\prime}({mx} )- m{\zeta _n}^{\prime}(x ){\Psi _n}({mx} )}}\\ {b_n} &= \frac{{m{\Psi _n}(x ){\Psi _n}^{\prime}({mx} )- {\Psi _n}^{\prime}(x ){\Psi _n}({mx} )}}{{m{\zeta _n}(x ){\Psi _n}^{\prime}({mx} )- {\zeta _n}^{\prime}(x ){\Psi _n}({mx} )}} \end{aligned}$$
where m is the ratio of the refractive index of the scattering particles to the refractive index of the surrounding medium.

Figure 8 shows the relationship between absorption coefficient and time, where the laser wavelength in Fig. 8(a) is 10.6 µm, and the laser wavelength in Fig. 8(b) is 3.8 µm. The results show that the effective absorption coefficient of aerosol varies with time on the order of microseconds, and the time has tended to be stable on the order of milliseconds.

 figure: Fig. 8.

Fig. 8. Absorption coefficient as a function of time. (a) λ = 10.6 µm, (b) λ = 3.8 µm.

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Conditions are used as the follows: the transmission distance of 5 km, the initial laser power of 50 kW, the wind speed of 5 m/s, and the laser radius of 0.2 m, the dynamic effect of the thermal blooming is simulated. Figure 9 shows the contour distributions of laser intensity at laser wavelengths of 10.6 µm and 3.8 µm, and the interception times are 1 µs, 10 us, 100 µs, and 1000 µs respectively. It can be seen from Fig. 9 that the laser intensity will change under the influence of thermal blooming at different times. The change of the laser wavelength of 3.8 µm is more obvious than that of the laser wavelength of 10.6 µm. This result is consistent with the trend of the absorption coefficient versus time plotted in Fig. 8. After the thermal blooming effect reaches the millisecond level, the laser intensity tends to be stable. Most of the studies on thermal blooming tend to be in a stable state.

 figure: Fig. 9.

Fig. 9. Dynamic simulation of thermal blooming at different wavelengths. (a1), (b1), (c1) and (d1) are results for λ = 10.6 µm. (a2), (b2), (c2) and (d2) are results for λ = 3.8 µm. The time t is changed from 1µs to 1000µs from the first column to the 4th one, respectively.

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4. Laboratory experiments

The liquid crystal thermal blooming simulator is based on LCOS from BNS Inc. Its resolution is 512 × 512, and the gray level is 256 steps and 8 bits. And the experimental optical path is shown in Fig. 10. The He-Ne laser with a wavelength of 632.8 nm is used as the light source. The simulated laser radius is 0.25 m, and the beam radius incident on the LCSLM is 2.5 mm. The scaling factor between simulated results and experimental ones of the object screen is 100. After being affected by the thermal halo effect, the image screen size of the simulated receiving beam is 2m × 2 m, the number of grids is 512 × 512, and the size of each pixel is about 3.9 mm. The actual size of the CCD is 18mm × 13.5 mm with 1440 × 1080 pixels, and the corresponding size of each pixel is 0.0125mm × 0.0125 mm. The scaling factor between simulated results and experimental results of the image screen is 312. The focal length of lens 1 and 2 are 20 mm and 400 mm, respectively. In this experiment, it is important for the phase extraction of thermal blooming distortion.

 figure: Fig. 10.

Fig. 10. Optical path of thermal blooming effect experiment based on LC-SLM

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Frederick G. Gebhardt [33] used PM for phase extraction, and expressed it as

$$\begin{array}{c} {\varphi _{BG}}({x,y,z} )={-} \frac{{\Delta {\varphi _G}}}{2}\textrm{exp} [{ - {{({y/a} )}^2}} ][{1 + erf({x/a} )} ]\\ \Delta {\varphi _G} = \left( {1/2\sqrt \pi } \right){N_D}\\ {N_D} = \frac{{4\sqrt 2 ({ - {n_T}} )k\alpha Pz}}{{\rho {c_p}vD}} \end{array}.$$

This paper proposes the phase extraction of PSM based on the basic principle of PSM introduced in section 2. PSM is to insert multiple phase screens between the object and the image. It uses the Fourier transform method between two adjacent screens to obtain the light field distribution on the second screen. Similarly, this method can obtain the final light field distribution on the CCD. The role of the medium in the whole propagation process is to generate thermal blooming distortion phase. In the calculation of a step size, the phase distortion can be expressed as

$$\begin{array}{c} \Delta \varphi = {{k\delta \bar{\varepsilon }\Delta z} / 2}\\ \delta \bar{\varepsilon } = \frac{1}{{\Delta z}}\int_{{z^n}}^{{z^n} + \Delta z} {\delta \varepsilon ({z^{\prime}} )dz^{\prime}} \\ \delta \varepsilon = 0.46{\rho _1}\\ {\rho _1} = \frac{{({\gamma - 1} )\alpha }}{{c_s^2v}}\int_{ - \infty }^x {I({x^{\prime},y,z} )dx^{\prime}} \end{array}.$$

The phase distortion of the whole propagation process is the result of each step accumulation. The initial parameter is set to: λ = 10.6µm, P = 50 kW, a = 0.2 m, α = 6.5 × 10−5 /m, v = 5 m/s. The transmission distances are 1 km, 3 km, 5 km and 10 km. The N values are 0.7047, 2.0192, 3.1581, and 5.6045, respectively. They are corresponding to experiments using the perturbation method, perturbation method, perturbation method, and phase screen method, respectively. The simulated phase and its corresponding experimental results using the formula method and the traditional phase extraction method are shown in Fig. 11, and the size of the pictures is 512 × 512. The second line shows that as the transmission distance increases, the thermal blooming distortion becomes stronger, and the beam gradually changes from a circular spot to a crescent shape. However, the experimental results obtained by using the traditional phase extraction method cannot see the obvious thermal blooming distortion effect.

 figure: Fig. 11.

Fig. 11. Thermal blooming phase and its corresponding laboratory results obtained by using formula method and traditional phase extraction method. (a1), (b1), (c1), (d1) are phase diagrams obtained by using the formula method (The unit of phase is rad.), and (a2), (b2), (c2), (d2) are its experimental results. (a3), (b3), (c3), (d3) are the phase diagrams obtained by the traditional phase extraction method, and (a4), (b4), (c4), (d4) are its experimental results.

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Table 1 records the transverse length of the beam and the position of the center of mass of the beam, and compares them with the results of numerical simulation. x1 and m1 represent the horizontal length of the beam and the abscissa of the centroid position in the numerical simulation results, and x2 and m2 represent the horizontal length of the beam and the abscissa of the centroid position in the experimental simulation results. It is worth noting that the length is represented by the number of pixels.

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Table 1. Experimental data recording table

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Calculate and analyze the data in Table 1 using the Pearson product-moment correlation coefficient. The formula is as follows:

$${\rho _{x,y}} = \frac{{{\mathop{\rm cov}} ({x,y} )}}{{{\sigma _x}{\sigma _y}}}.$$

Substituting x1 and x2 into formula (20), the result is 0.9858. Substituting m1 and m2 into formula (20), the result is 0.9896. The absolute value of the correlation coefficient is very close to 1. This proves that the laboratory experimental results are very close to the numerical simulation results.

After validating this method, we use liquid crystal spatial light modulator to emulate experimentally the dynamic effect of thermal blooming, and the results are shown in Fig. 12. Conditions are used as the follows: the transmission distance of 5 km, the initial laser power of 50 kW, the wind speed of 5 m/s, and the laser radius of 0.2 m. The results indicate how the laser beam with a wavelength of 3.8 µm is affected by the thermal blooming over time when it propagates in the atmosphere. It can be seen from Fig. 12 that this variation trend is similar to the simulation results and coincides with the variation of the aerosol effective absorption coefficient over time. Reference [44] conducted thermal blooming experiments by building a thermal blooming cell, and the experimental results showed that the thermal blooming distortion change was completed in the order of microseconds, which also verified that the experimental results in Fig. 12 were reasonable.

 figure: Fig. 12.

Fig. 12. Dynamic thermal blooming experiment results, (a)1µs, (b)10µs, (c)100µs, (d)1000µs.

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5. Conclusion

This paper gives the most suitable range for both PM and PSM according to the generalized distortion parameter N based on experimental results. For N less than 4.8, the error generated by PM is smaller, and for N larger than 4.8, the error generated by PSM is smaller. The error of the simulation results relative to the reference line is within 0.2. In addition, this paper analyzes the time-related quantities among the various factors affecting thermal blooming, and realizes the dynamic simulation of thermal blooming by using the variation of absorption coefficient with time. LC-SLM was used to emulate experimentally the thermal blooming effect with accurate phase extraction. Different from the traditional method of phase extraction by using the argument angle function, this paper extracts the phase based on the basic principle of the numerical simulation method, which will avoid the calculation error when directly performing the argument angle function on the light field distribution of the receiving laser to extract the phase. This makes the final experimental simulation results more consistent with the numerical simulation results, which is validated by the experimental results.

Funding

National Natural Science Foundation of China (No. 61475152, No. 62205127); the Fund for Key Laboratory of Electro-Optical Countermeasures Test & Evaluation Technology (GKCP2021001).

Disclosures

The authors declare no conflicts of interest in this paper.

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.

References

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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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Figures (12)
Fig. 1.
Fig. 1. Contour diagrams of light intensity simulated by the two methods at different transmission distances. (a1), (b1), (c1) and (d1) are the results of PM. (a2), (b2), (c2) and (d2) are the results of PSM. The transmission distance increases from 1 km to 10 km from first column to the 4th one, respectively.

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Fig. 2.
Fig. 2. Contour diagrams of light intensity under different initial laser powers simulated by the two methods. (a1), (b1), (c1) and (d1) are the results of PM. (a2), (b2), (c2) and (d2) are the results of PSM. The initial laser power increases from 10 kW to 100 kW from the first column to the 4th one, respectively.

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Fig. 3.
Fig. 3. Contour maps of light intensity at different wind speeds simulated by the two methods. (a1), (b1), (c1) and (d1) are the simulation results of the perturbation method. (a2), (b2), (c2) and (d2) are the simulation results of PSM. The wind speed is ranged from 2 m/s to 10 m/s from the first column to the 4th one, respectively.

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Fig. 4.
Fig. 4. Blooming IREL dependence on the parameter N for experimental and simulated results. (Reference [33], Fig. 6)

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Fig. 5.
Fig. 5. (a) normalized peak intensity as a function of N. perturbation: PM; phase screen: PSM. (b) the errors of normalized peak intensity as a function of N at different transmission distances for PM and PSM.

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Fig. 6.
Fig. 6. (a) Normalized peak intensity as a function of N for PM and PSM at different initial laser powers. perturbation: PM; phase screen: PSM. (b) the errors of normalized peak intensity as a function of N at different initial laser powers.

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Fig. 7.
Fig. 7. (a)Normalized peak intensity as a function of N for PM and PSM at different wind speeds. perturbation: PM; phase screen: PSM. (b) is the error of normalized peak intensity as a function of N at different wind speeds.

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Fig. 8.
Fig. 8. Absorption coefficient as a function of time. (a) λ = 10.6 µm, (b) λ = 3.8 µm.

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Fig. 9.
Fig. 9. Dynamic simulation of thermal blooming at different wavelengths. (a1), (b1), (c1) and (d1) are results for λ = 10.6 µm. (a2), (b2), (c2) and (d2) are results for λ = 3.8 µm. The time t is changed from 1µs to 1000µs from the first column to the 4th one, respectively.

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Fig. 10.
Fig. 10. Optical path of thermal blooming effect experiment based on LC-SLM

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Fig. 11.
Fig. 11. Thermal blooming phase and its corresponding laboratory results obtained by using formula method and traditional phase extraction method. (a1), (b1), (c1), (d1) are phase diagrams obtained by using the formula method (The unit of phase is rad.), and (a2), (b2), (c2), (d2) are its experimental results. (a3), (b3), (c3), (d3) are the phase diagrams obtained by the traditional phase extraction method, and (a4), (b4), (c4), (d4) are its experimental results.

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Fig. 12.
Fig. 12. Dynamic thermal blooming experiment results, (a)1µs, (b)10µs, (c)100µs, (d)1000µs.

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Tables (1)

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Table 1. Experimental data recording table

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Equations (21)

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I ( x , y , z ) I u ( x , y , z ) e − α z = exp { − N c [ 2 ( x a ) e [ − ( x 2 + y 2 ) / a 2 ] + π 2 e ( − y 2 / a 2 ) ( 1 − 4 y 2 a 2 ) ( 1 + e r f ( x a ) ) ] } N c = ( − 2 d n / d T I 0 z n 0 ρ C p v a ) [ 1 − ( 1 − e − α z ) α z ] ≈ ( − d n / d T I 0 z 2 α n 0 ρ C p v a ) , α z ≪ 1
2 i k ∂ φ ∂ z = ∇ T 2 φ + k 2 [ n 2 ( x , y , z , t ) n 0 2 − 1 ] φ .
φ n + 1 = φ n exp [ i 2 k ( ∇ T 2 + k 2 δ ε ¯ ) Δ z ]
φ ( x , y , z ) = ∑ m = − N 2 N 2 − 1 ∑ n = − N 2 N 2 − 1 φ m n ( z ) exp [ i 2 π ( m x + n y ) L ] .
φ m n ( z , t ) = 1 N 2 ∑ j = 0 N − 1 ∑ i = 0 N − 1 φ ( j Δ x , i Δ y , z ) exp [ − i 2 π ( m j + n i ) N ] .
φ m n ( z + Δ z 2 , t ) = φ m n ( z , t ) exp [ − i 2 k ( 2 π L ) 2 ( m 2 + n 2 ) Δ z 2 ] .
φ ( j Δ x , i Δ y , z + Δ z 2 , t ) = ∑ m = − N 2 N 2 − 1 ∑ n = − N 2 N 2 − 1 φ m n ( z + Δ z 2 , t ) exp [ − i 2 π ( m j + n i ) N ] .
φ ′ ( j Δ x , i Δ y , z + Δ z 2 , t ) = φ ( j Δ x , i Δ y , z + Δ z 2 , t ) exp ( i k 2 δ ε ¯ Δ z ) .
φ m n ′ ( z , t ) = 1 N 2 ∑ j = 0 N − 1 ∑ i = 0 N − 1 φ ′ ( j Δ x , i Δ y , z + Δ z 2 , t ) exp [ − i 2 π ( m j + n i ) N ] .
φ m n ′ ( z + Δ z , t ) = φ m n ′ ( z , t ) exp [ − i 2 k ( 2 π L ) 2 ( m 2 + n 2 ) Δ z 2 ] .
φ ′ ( j Δ x , i Δ y , z + Δ z , t ) = ∑ m = − N 2 N 2 − 1 ∑ n = − N 2 N 2 − 1 φ m n ′ ( z + Δ z , t ) exp [ − i 2 π ( m j + n i ) N ] .
I R E L = 1 1 + 0.0625 N 2 .
N = N c f ( N E ) q ( N F ) s ( N ω ) q ( N F ) = [ 2 N F 2 N F − 1 ] [ 1 − I n N F N F − 1 ] f ( N E ) = 2 N E 2 [ N E − 1 + exp ( − N E ) ] s ( N ω ) = 2 N ω 2 [ ( N ω + 1 ) I n ( N ω + 1 ) − N ω ]
N c = 16 2 ( − n T ) α P L 2 π n 0 ρ c p v D 3 N E = ( α + s ) z N F = k a 2 / z N ω = ω z v 0 .
I e r r o r 1 = I P M − I E M I e r r o r 2 = I P S M − I E M .
α e f f ( t ) = ∫ Q a b s ( a ) π a 2 n ( a ) { 1 − exp ( − t τ ( a ) ) } d a n ( a ) = { 2 × 10 − 12 a − 4 , 2 × 10 − 2 < a < 30 μ m 0 , e l s e w h e r e τ ( a ) = a 2 ρ p c p 3 k a .
Q a b s = σ a b s π r 2 = Q e x t + Q s Q e x t = 2 x 2 ∑ n = 1 ∞ ( 2 n + 1 ) R e ( a n + b n ) Q s = 2 x 2 ∑ n = 1 ∞ ( 2 n + 1 ) ( | a n | 2 + | b n | 2 )
a n = Ψ n ( x ) Ψ n ′ ( m x ) − m Ψ n ′ ( x ) Ψ n ( m x ) ζ n ( x ) Ψ n ′ ( m x ) − m ζ n ′ ( x ) Ψ n ( m x ) b n = m Ψ n ( x ) Ψ n ′ ( m x ) − Ψ n ′ ( x ) Ψ n ( m x ) m ζ n ( x ) Ψ n ′ ( m x ) − ζ n ′ ( x ) Ψ n ( m x )
φ B G ( x , y , z ) = − Δ φ G 2 exp [ − ( y / a ) 2 ] [ 1 + e r f ( x / a ) ] Δ φ G = ( 1 / 2 π ) N D N D = 4 2 ( − n T ) k α P z ρ c p v D .
Δ φ = k δ ε ¯ Δ z / 2 δ ε ¯ = 1 Δ z ∫ z n z n + Δ z δ ε ( z ′ ) d z ′ δ ε = 0.46 ρ 1 ρ 1 = ( γ − 1 ) α c s 2 v ∫ − ∞ x I ( x ′ , y , z ) d x ′ .
ρ x , y = cov ( x , y ) σ x σ y .
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