Characteristics of high-power partially coherent laser beams propagating upwards in the turbulent atmosphere

Yu Deng; Huan Wang; Xiaoling Ji; Xiaoling Ji; Xiaoling Ji; Xiaoqing Li; Hong Yu; Lifeng Chen

doi:10.1364/OE.399401

1. Introduction

It is important to study the characteristics of high-power laser beams propagating upwards in the turbulent atmosphere for laser ablation propulsion’s applications in space, including space-debris removal and launching small probes into Low Earth Orbit, etc [1]. The nonlinear self-focusing effect on beam propagation must be considered because the laser power is well above the critical power for self-focusing in air [2,3]. Recently, some studies were carried out concerning the self-focusing effect of laser beams in the inhomogeneous atmosphere. In 2009, Rubenchik et al. indicated that the laser beam can be compressed as a whole and the beam collapse can be avoided because the self-focusing length exceeds the atmosphere thickness even if the laser power is much larger than the critical value [4]. In 2014, Rubenchik et al. demonstrated the self-focusing can noticeably decrease the laser intensity on the target, and the detrimental effect can be compensated for by applying the initial beam defocusing [3]. In 2018, we showed that the temporal pulse splitting may appear on the space debris due to the interplay of the group-velocity dispersion effect and the self-focusing effect in the inhomogeneous atmosphere [5]. In 2019, we found that the uniform irradiation on the space debris is achieved because of the phase modulation caused by self-focusing in the inhomogeneous atmosphere [6], and studied the effect of spatial coherence on laser space-debris removal in the inhomogeneous atmosphere [7]. However, the effect of atmospheric turbulence on self-focusing wasn’t considered in these studies [3–7].

It is known that the laser beam quality will also be affected by atmospheric turbulence. Many studies were carried out concerning the propagation characteristics of laser beams in atmospheric turbulence [8–13], in which the laser power was very low and the self-focusing effect wasn’t considered. In 2017, Peñano et al. studied the self-channeling of high-power laser pulses through strong atmospheric turbulence, and found that the pulse can propagate many Rayleigh lengths when the laser power is close to the self-focusing critical power of air and the transverse dimensions of the pulse are smaller than the coherence diameter of turbulence [14]. In 2017, Hafizi et al. studied the laser beam self-focusing in turbulent dissipative media by applying a method of moments, and found that dissipation reduced the self-focusing and led to chromatic aberration [15]. However, only the propagation of laser beams in the homogeneous atmosphere within several kilometers was examined in Refs. [14,15].

Usually, the nonlinear Schrödinger (NLS) equation is applied to describe the laser beam propagation in nonlinear media. But it is very difficult to obtain the analytical solution of NLS equation, except for several special cases [16–19]. In general, the NLS equation is solved numerically by using the multi-phase screen method [3–6,20]. In 2016, Vaseva et al. proposed the “thin window” (TW) model to study the propagation of high-power laser beams in the inhomogeneous atmosphere [21]. But the analytical expression of the intensity wasn’t derived, and only semi-analytical expression of the beam width was obtained in Ref. [21]. In addition, the atmospheric turbulence wasn’t considered, and only fully coherent laser beams were considered in Ref. [21]. In this paper, the characteristics of high-power partially coherent laser beams propagating upwards in the turbulent atmosphere are studied, where the principal features of diffraction, nonlinear self-focusing and turbulence are considered. Based on the TW model, the analytical propagation formulae (e.g., the cross-spectral density function, the intensity, the beam width and the coherence width) are derived by using the quadratic approximation of nonlinear phase shift, which are confirmed by using the numerical results. The influence of laser elevation on self-focusing, turbulence and diffraction effects is investigated in detail. Furthermore, the optimization of the optics system and laser to decrease the beam spot size on the target is also examined. The analytical formulae of fully coherent laser beams can be treated as a special case of our results. The results obtained in this paper are theoretical and practical interest.

2. Theoretical model and analytical formulae

For simplicity, we assume that a partially coherent laser beam propagates vertically upwards in the turbulent atmosphere (see Fig. 1). It is demonstrated that the TW approximation is quite reliable because the thickness of the atmosphere is much smaller than the focusing length [3]. According to the TW model, propagation through the atmosphere results in phase distortion only [21]. Furthermore, we demonstrated that the phase modulation caused by the nonlinear self-focusing in the inhomogeneous atmosphere nearly doesn’t affect the intensity distribution within the atmospheric thickness [6]. Thus, the nonlinear phase shift scale can be expressed as [21]

(1)$$\phi = k\int_{{h_1}}^{{z_1}} {{n_2}} (z ){I_0}(r )\textrm{d}z = k{n_{20}}{h_0}\frac{{2P}}{{\pi {w_0}^2}}\left[ {\textrm{exp}\left( { - \frac{{{h_1}}}{{{h_0}}}} \right) - \textrm{exp}\left( { - \frac{{{z_1}}}{{{h_0}}}} \right)} \right]\exp \left( { - \frac{{2{r^2}}}{{w_0^2}}} \right),$$

where ${n_2}(z )= {n_{20}}\exp ({{{ - z} / {{h_0}}}} )$ is the nonlinear refractive index, n₂₀ is the nonlinear refractive index on the ground, and h₀= 6km is a constant; I₀(r)=[ 2P/(πw₀²)]exp(−2r²/ w₀²) is the initial intensity, P is the beam power, and w₀ is the mean square beam width; h₁ is the laser elevation (e.g., h₁=5.9km if the laser is sited at the Uhuru site on Kilimanjaro [22], and h₁=0 if the laser is sited on the ground); z₁=20km is the atmospheric thickness; k=2π/λ is the wave-number, and λ is the wavelength.

Fig. 1. Diagram of the laser beam propagation in the turbulent atmosphere.

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It is known that the cross-spectral density function is very useful to handling partially coherent laser beams. The Gaussian Schell-mode (GSM) beam is usually adopted as a typical example of partially coherent laser beams. For example, the GSM beam is adopted to describe the propagation of high-power partially coherent fiber laser beams in a real environment [23]. The initial cross-spectral density function of GSM beams at z = h₁ plane is written as

(2)$$\begin{aligned}{c} W({{r_{01}},{r_{02}},{h_1}} )&= \frac{{2P}}{{\pi {w_0}^2}}\exp \left( { - \frac{{r_{01}^2 + r_{02}^2}}{{w_0^2}}} \right)\exp \left[ { - \frac{{{{({{r_{01}} - {r_{02}}} )}^2}}}{{2\sigma_0^2}}} \right]\\ &\quad\times \exp \left[ { - \textrm{i}\frac{{k({r_{01}^2 - r_{02}^2} )}}{{2f}}} \right]\exp [{ - \textrm{i}({{\phi_1} - {\phi_2}} )} ], \end{aligned}$$

where σ₀ is the initial spatial coherence width, f is the focal length of a lens located at z = h₁ plane.

To obtain the analytical results, a quadratic approximation of nonlinear phase shift is adopted in this paper, i.e., the last term in Eq. (1) is approximated by

(3)$$\exp \left( { - \frac{{2{r^2}}}{{w_0^2}}} \right) \approx \left( {1 + {\xi_1}\frac{r}{{{w_0}}} + {\xi_2}\frac{{{r^2}}}{{w_0^2}}} \right),$$

where ${\xi _1} ={-} 0.3916$ and ${\xi _2} ={-} 0.7004$. The validity of this approximation is proved by us (see Fig. 2). Figure 2 shows that, within the main intensity range, the low-order Taylor expansions (e.g., second-order and third-order Taylor expansions) can’t fit the exponential function, but the quadratic function can do. It is noted that the analytical results can’t be generated if high-order Taylor expansions are adopted. In particular, ${\xi _1} ={-} 0.3916$ and ${\xi _2} ={-} 0.7004$ obtained in this paper are robust if the initial intensity is the Gaussian profile.

Fig. 2. Confirmation of the validity of the approximation of Eq. (3).

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Based on the quadratic approximation of exponential function (i.e., Eq. (3)), Eq. (2) can be simplified as

(4)$$\begin{aligned} W({{r_1},{r_2},{h_1}} )&= \frac{{2P}}{{\pi w_0^2}}\exp \left( { - \frac{{r_1^2 + r_2^2}}{{w_0^2}}} \right)\exp \left[ { - \frac{{{{({{r_1} - {r_2}} )}^2}}}{{2\sigma_0^2}}} \right]\\ &\quad\times \exp \left[ { - \textrm{i}\frac{{k({r_1^2 - r_2^2} )}}{{2{R_0}}}} \right]\exp \left[ { - \textrm{i}\frac{{2Pk{n_{20}}{h_0}{\xi_1}({{r_1} - {r_2}} )}}{{\pi w_0^3}}} \right], \end{aligned}$$

where ${1 / {{R_0} = {1 / f}}} + {1 / {{f_\textrm{s}}}}$, and

(5)$${{{f_\textrm{s}} = \pi w_0^4} / {4{\xi _2}{n_{20}}P{h_0}[{\textrm{exp}({ - {{{h_1}} / {{h_0}}}} )- \textrm{exp}({ - {{{z_1}} / {{h_0}}}} )} ]}}.$$

The f_s denotes the focal length of self-focusing effect.

In this paper, we concentrate on the principal features of diffraction, self-focusing and turbulence. Based on the generalized Huygens-Fresnel principle, the cross-spectral density function of GSM beams propagating through the turbulent atmosphere is expressed as

(6)$$\begin{aligned} W({{r_1},{r_2},z} )&= {\left[ {\frac{k}{{2\pi ({z - {h_1}} )}}} \right]^2}\int\!\!\!\int {{\textrm{d}^2}{r_{01}}} \int\!\!\!\int {{\textrm{d}^2}{r_{02}}W({{r_{01}},{r_{02}},{h_1}} )} \\ &\quad\times \exp \left\{ { - \frac{{\textrm{i}k}}{{2({z - {h_1}} )}}[{{{({{r_1} - {r_{01}}} )}^2} - {{({{r_2} - {r_{02}}} )}^2}} ]} \right\}{\left\langle {\exp [{\psi ({{r_{01}},{r_1}} )+ {\psi^\ast }({{r_{02}},{r_2}} )} ]} \right\rangle _\textrm{m}}, \end{aligned}$$

where Ψ is the phase function that depends on the properties of the medium, and < >_m denotes the average over the ensemble of the turbulent medium. Based on the quadratic approximation of the Rytov’s phase structure function, the ensemble average term in Eq. (6) can be expressed approximately as [24]

(7)$${\left\langle {\exp [{\psi ({{r_{01}},{r_1}} )+ {\psi^\ast }({{r_{02}},{r_2}} )} ]} \right\rangle _\textrm{m}} \approx \exp \left\{ { - \frac{1}{{\rho_0^2}}[{{{({{r_{01}} - {r_{02}}} )}^2} + ({{r_{01}} - {r_{02}}} )({{r_1} - {r_2}} )+ {{({{r_1} - {r_2}} )}^2}} ]} \right\},$$

where ${\rho _0} = {\left\{ {\textrm{1}\textrm{.46}{k^2}\int_{{h_1}}^z {C_\textrm{n}^2} {{[{\textrm{1} - {h / {({z - {h_1}} )}}} ]}^{{5 / 3}}}\textrm{d}h} \right\}^{{{ - 3} / 5}}}$ is the spatial coherence radius of a spherical wave propagating in turbulence, and $C_\textrm{n}^\textrm{2}$ is the atmospheric structure constant of refractive index. The smaller value of ρ₀ means the influence of turbulence on the beam quality becomes stronger. In this paper, we adopt the ITU-R model presented in 2001 to describe the model of the altitude dependent structure constant, i.e. [25],

(8)$$C_\textrm{n}^\textrm{2}(h )= 8.148 \times {10^{ - 56}}{V^2}{h^{10}}\exp ({ - {h / {1000}}} )+ 2.7 \times {10^{ - 16}}\exp ({ - {h / {1500}}} )+ {C_0}\exp ({ - {h / {100}}} ),$$

where $V = {({v_\textrm{g}^2 + 30.69{v_g} + 348.91} )^{{1 / 2}}}$ is the wind speed along the vertical path, and v_g is the ground wind speed (setting v_g=2.8 m/s in this paper), C₀ is the nominal value of ground level. The turbulence effect becomes stronger as C₀ increases.

Introducing variables

(9)$${u_0} = \frac{{{r_{01}} + {r_{02}}}}{2},{v_0} = {r_{02}} - {r_{01}},$$

(10)$$u = \frac{{{r_1} + {r_2}}}{2},v = {r_2} - {r_1},$$

Equation (4) can be rewritten as

(11)$$\begin{aligned} W({{u_0},{v_0},{h_1}} )&= \frac{{2P}}{{\pi w_0^2}}\exp \left( { - \frac{{2u_0^2}}{{w_0^2}}} \right)\exp \left( { - \frac{{v_0^2}}{{2w_0^2}}} \right)\exp \left( { - \frac{{v_0^2}}{{2\sigma_0^2}}} \right)\\ &\quad \times \exp \left( { - \frac{{\textrm{i}k{u_0}{v_0}}}{{{R_0}}}} \right)\exp \left( {\textrm{i}\frac{{2{\xi_1}kP{n_{20}}{h_0}{v_0}}}{{\pi w_0^3}}} \right), \end{aligned}$$

and Eq. (6) can be rewritten as

(12)$$\begin{aligned} W({u,v,z} )&= {\left[ {\frac{k}{{2\pi ({z - {h_1}} )}}} \right]^2}\int\!\!\!\int {{\textrm{d}^2}{u_0}} \int\!\!\!\int {{\textrm{d}^2}{v_0}W({{u_0},{v_0},{h_1}} )} \exp \left[ { - \frac{{\textrm{i}k}}{{({z - {h_1}} )}}({{v_0} - v} )({u - {u_0}} )} \right]\\ &\quad\times \exp \left[ { - \frac{1}{{\rho_0^2}}({v_0^2 + {v_0}v + {v^2}} )} \right]. \end{aligned}$$

On substituting from Eq. (11) into Eq. (12), and by using the integral formula $\int {\exp ({ - a{x^2} + bx} )\textrm{d}x = } \frac{{\sqrt \pi }}{{\sqrt a }}\exp \left( {\frac{{{b^2}}}{{4a}}} \right)$, we can obtain the cross-spectral density function of GSM beams propagating through the turbulent atmosphere, i.e.,

(13)$$\begin{aligned} W({u,v,z} )&= \frac{{2P}}{{\pi w_0^2}}{\left[ {\frac{k}{{2\pi ({z - {h_1}} )}}} \right]^2}\frac{{\pi w_0^2}}{2}\frac{\pi }{\alpha }\exp \left( {\frac{{\textrm{i}k}}{{z - {h_1}}}vu} \right)\exp \left( { - \frac{1}{{\rho_0^2}}{v^2}} \right)\exp \left[ { - \frac{{{k^2}w_0^2}}{{8{{({z - {h_1}} )}^2}}}{v^2}} \right]\\ &\quad\times \exp \left( {\frac{{{\beta^2}}}{{4\alpha }}{v^2}} \right)\exp \left[ { - \frac{{{k^2}}}{{4\alpha {{({z - {h_1}} )}^2}}}{u^2}} \right]\exp \left[ { - \frac{{ik\beta }}{{2\alpha ({z - {h_1}} )}}uv} \right]\\ &\quad\times \exp \left( { - \frac{{{k^2}{n_{20}}^2{h_0}^2{P^2}{\xi_1}^2}}{{\alpha {\pi^2}w_0^6}}} \right)\exp \left( { - \textrm{i}\frac{{{\xi_1}k{n_{20}}{h_0}{C_1}P\beta v}}{{\alpha \pi w_{_0}^3}}} \right)\exp \left[ { - \frac{{{\xi_1}{k^2}{n_{20}}{h_0}Pu}}{{\alpha \pi w_{_0}^3({z - {h_1}} )}}} \right], \end{aligned}$$

where

(14)$$\alpha = \frac{{{k^2}w_0^2}}{{8{{({z - {h_1}} )}^2}}}{\left( {1 - \frac{{z - {h_1}}}{{{R_0}}}} \right)^2} + \frac{1}{{{\varepsilon ^2}}},$$

(15)$$\beta = \frac{{{k^2}w_0^2}}{{4{{({z - {h_1}} )}^2}}}\left( {1 - \frac{{z - {h_1}}}{{{R_0}}}} \right) + \frac{1}{{\rho _0^2}},$$

(16)$$\frac{1}{{{\varepsilon ^2}}} = \frac{1}{{2{w_0}^2}}\textrm{ + }\frac{1}{{2{\sigma _0}^2}} + \frac{1}{{{\rho _0}^2}}.$$

On substituting from Eq. (10) into Eq. (13), Eq. (13) can be rewritten as

(17)$$\begin{aligned} W({{r_1},{r_2},z} )&= \frac{{2P}}{{\pi w_0^2}}\frac{{{k^2}w_0^2}}{{8\alpha {{({z - {h_1}} )}^2}}}\exp \left( { - \frac{{{\xi_1}^2{k^2}{n_{20}}^2{P^2}{h^2}}}{{\alpha {\pi^2}w_0^6}}} \right)\exp \left[ { - \frac{{{k^2}({r_1^2 + r_2^2} )}}{{8\alpha {{({z - {h_1}} )}^2}}}} \right]\\ &\quad\times \exp \left\{ {\left[ {\frac{{{k^2}}}{{16\alpha {{({z - {h_1}} )}^2}}} - \frac{1}{{\rho_0^2}} - \frac{{{k^2}w_0^2}}{{8{{({z - {h_1}} )}^2}}} + \frac{{{\beta^2}}}{{4\alpha }}} \right]{{({{r_1} - {r_2}} )}^2}} \right\}\\ &\quad\times \exp \left\{ {\left[ {\frac{\beta }{{2\alpha ({z - {h_1}} )}} - \frac{1}{{z - {h_1}}}} \right]\frac{{\textrm{i}k({r_1^2 - r_2^2} )}}{2}} \right\}\exp \left[ {\textrm{i}\frac{{{\xi_1}k{n_{20}}{h_0}P\beta ({{r_1} - {r_2}} )}}{{\alpha \pi w_0^3}}} \right]\\ &\quad\times \exp \left[ { - \frac{{{\xi_1}{k^2}{n_{20}}{h_0}P}}{{\alpha \pi w_0^3({z - {h_1}} )}}({{r_1} + {r_2}} )} \right]. \end{aligned}$$

Letting r₁=r₂ in Eq. (17), we can obtain the intensity of GSM beams propagating through turbulent atmosphere, i.e.,

(18)$$\begin{aligned} I({r,z} )&= \frac{{2P}}{{\pi w_0^2}}\frac{{{k^2}w_0^2}}{{8\alpha {{({z - {h_1}} )}^2}}}\exp \left( { - \frac{{{\xi_1}^2{k^2}{n_{20}}^2{h_0}^2{P^2}}}{{\alpha {\pi^2}w_0^6}}} \right)\\ &\quad\times \exp \left[ { - \frac{{{k^2}{r^2}}}{{4\alpha {{({z - {h_1}} )}^2}}}} \right]\exp \left[ { - \frac{{2{\xi_1}{k^2}{n_{20}}{h_0}Pr}}{{\alpha \pi w_0^3({z - {h_1}} )}}} \right]. \end{aligned}$$

In Eq. (18), we have $\frac{{{\xi _1}^2{k^2}{n_{20}}^2{h_0}^2{P^2}}}{{\alpha {\pi ^2}w_0^6}} \ll 1$ and $\frac{{2{\xi _1}{k^2}{n_{20}}{h_0}Pr}}{{\alpha \pi w_{_0}^3({z - {h_1}} )}} \ll 1$. Thus, the first and the third exponential terms in Eq. (18) can be omitted. Based on this approximation and the definition of the mean square beam width, we obtain the beam width of GSM beams propagating through the turbulent atmosphere, i.e.,

(19)$$\begin{aligned}{w^2}(z )&= \frac{{2\int\!\!\!\int {{r^2}I({r,z} ){\textrm{d}^2}r} }}{{\int\!\!\!\int {I({r,z} ){\textrm{d}^2}r} }} \approx \frac{{8\alpha {{({z - {h_1}} )}^2}}}{{{k^2}}}\\ &\textrm{ = }w_0^2 - w_0^2\left[ {2\frac{{({z - {h_1}} )}}{{{R_0}}} - {{\left( {\frac{{z - {h_1}}}{{{R_0}}}} \right)}^2}} \right] + \left( {\frac{4}{{w_0^2}} + \frac{4}{{\sigma_0^2}} + \frac{8}{{\rho_0^2}}} \right){\left( {\frac{{z - {h_1}}}{k}} \right)^2}. \end{aligned}$$

It is noted that the expression of α (i.e. Equation (14)) is applied to obtain Eq. (19). On the right side of Eq. (19), the first term is the initial beam width, the second term represents the beam focusing due to the lens and the nonlinear self-focusing effect, and other terms denote the beam spreading due to the beam diffraction, the beam spatial coherence and the atmospheric turbulence in turn.

A quantitative measure of the strength of the field correlation is given by the spectral degree of coherence, which is defined as [26]

(20)$$\mu ({{r_1},{r_2},z} )= \frac{{W({{r_1},{r_2},z} )}}{{\sqrt {W({{r_1},{r_1},z} )} \sqrt {W({{r_2},{r_2},z} )} }}.$$

On the other hand, the spectral degree of coherence of GSM beams can also be expressed as

(21)$$\mu ({{r_1},{r_2},z} )= \exp \left[ { - \frac{{{{({{r_1} - {r_2}} )}^2}}}{{2{\sigma^2}(z )}}} \right].$$

Substituting from Eq. (17) into Eq. (20), and considering Eq. (21), we obtain the coherence width σ(z) of GSM beams propagating through the turbulent atmosphere, i.e.,

(22)$$\frac{1}{{{\sigma ^2}(z )}} ={-} \frac{{{k^2}}}{{8\alpha {{({z - {h_1}} )}^2}}} - \frac{{{\beta ^2}}}{{2\alpha }} + \frac{2}{{\rho _0^2}} + \frac{{{k^2}w_0^2}}{{4{{({z - {h_1}} )}^2}}}.$$

From Eq. (22) together with Eqs. (14)–(16), one can conclude that the coherence width decreases due to turbulence and self-focusing effects.

The validity of Eqs. (19) and (22) depends on the quadratic approximation of the nonlinear phase shift (see Eq. (3)) and the quadratic approximation of the Rytov’s phase structure function (see Eq. (7)). The validity of Eq. (3) is proved by us (see Fig. 1), and the validity of Eq. (7) was verified by Leader [27]. Therefore, Eqs. (19) and (22) obtained in this paper are valid.

On the other hand, we confirm numerically the validity of Eq. (19) by using the multi-phase screen approach to deal with the nonlinear self-focusing. In this paper, unless specified, the calculation parameters are taken as λ=1.06 μm, w₀=1 m, n₂₀=4.2×10⁻¹⁹ cm²/W. The Gaussian beam critical power P_crGs=λ²/(2πn₀n₂₀) is adopted as the normalized factor to the beam power. In addition, the target altitude is adopted as z=500 km, and lens focal length f=500-h₁ km. Figure 3 shows the beam width w_tar on the target versus the relative power P/P_crGs, where the results obtained by using both Eq. (19) and numerical simulation are given. One can see that the result obtained by using Eq. (19) is in consistent with that by using numerical simulation method when the beam power is not very high. However, the error of Eq. (19) increases as the beam power P increases (i.e., the self-focusing effect becomes stronger). For example, the relative errors are 1.35%, 2.08%, 4.09% and 6.30% when P/P_cr= 1000, 3000, 4000 and 5000, respectively.

Fig. 3. Confirmation of the validity of Eq. (19). Beam width w_tar on the target versus the relative power P/P_crGs, h₁=0, σ₀→∞, ρ₀→∞.

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It is noted that the integral range of beam width with respect to r/w₀ in Fig. 3 is [0, ∞]. Because the intensity concentrates near the z axis, the result obtained by using Eq. (19) is in consistent with that by using numerical simulation when the beam power is not very high. However, the beam spot size on the target increases due to self-focusing effect. This is the physical reason why the error of Eq. (19) increases when the beam power is very high.

3. Changes of beam propagation characterizations

The changes of beam width w and the spatial coherence width σ for different initial spatial coherence width σ₀ versus the propagation distance z are shown in Figs. 4(a) and 4(b), respectively. As the σ₀ increases, the w decreases (see Fig. 4(a)), and σ increases (see Fig. 4(b)). Furthermore, there exists a minimum of σ versus z (see Fig. 4(b)). The self-focusing and turbulence effects cause a decrease of σ, but the diffraction causes an increase of σ. This is the physical reason why a minimum of σ appears as the propagation distance z increases.

Fig. 4. (a) the beam width w, and (b) the spatial coherence width σ versus the propagation distance z, P=1000P_crGs, C₀=1.7×10⁻¹⁶ m^-2/3, h₁=0.

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The changes of beam width w versus the propagation distance z for different effects are shown in Fig. 5. In comparison with the linear propagation in free space (see the green line), in the atmosphere the position of the beam width minimum is further away from the target due to self-focusing effect (see the blue line). However, this situation changes, and the beam spreads greatly due to turbulence effect (see the black line). In addition, compared the red line with the black line, one can see that the turbulence is a significant effect on beam propagation characterization.

Fig. 5. Beam width w versus the propagation distance z for different effects, P=1000P_crGs, C₀=1.7×10⁻¹⁶ m^-2/3, σ₀=1 m, h₁=0.

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4. Influence of laser elevation on the self-focusing, turbulence and diffraction effects

It is known that, both the nonlinear refractive index n₂ and the atmospheric structure constant $C_\textrm{n}^\textrm{2}$ of refractive index are the functions of altitude. Therefore, the self-focusing and turbulence effects depend on the laser elevation. The changes of beam width w versus the propagation distance z for different laser elevation h₁ are shown in Fig. 6. One can see that w decreases as the h₁ increases. The physical reason is that the self-focusing and turbulence effects become weaker as the h₁ increases.

Fig. 6. Beam width w versus the propagation distance z for different laser elevation h₁, P=1000P_crGs, C₀=1.7×10⁻¹⁶ m^-2/3, σ₀=1 m.

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The influence of laser elevation on self-focusing and turbulence effects, which is stronger? The changes of beam width w_tar on the target versus P/P_crGs and log(C₀) are shown in Fig. 7. When h₁=0, the influence of turbulence on w_tar is stronger than that of self-focusing (see Fig. 7(a)). However, the opposite situation appears when h₁=6km (see Fig. 7(b)). It means that, the turbulence effect plays the main role when the laser is sited on the ground, while the self-focusing effect plays the main role when the laser is placed at a high elevation. It is known that the self-focusing effect is suppressed by the turbulence effect because of beam spreading. On the other hand, the self-focusing and turbulence effects become weaker as the laser elevation increases. Therefore, the influence of turbulence on the self-focusing decreases as the laser elevation increases. This is the physical reason why the influence of laser elevation on turbulence effect is stronger than that on self-focusing effect.

Fig. 7. Beam width w_tar on the target versus P/P_crGs and log(C₀), σ₀=3 m. (a) h₁=0; (b) h₁=6 km.

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It is known that the diffraction depends on the beam spatial coherence. The changes of beam width w_tar on the target versus log(C₀) and σ₀ are shown in Fig. 8, and those versus P/P_crGs and σ₀ are shown in Fig. 9. Figure 8(a) indicates that the influence of turbulence on w_tar is stronger than that of spatial coherence (i.e., diffraction) when h₁=0. However, the opposite situation appears when h₁=6km (see Fig. 8(b)). It implies that influence of laser elevation on turbulence effect is stronger than that on diffraction effect. On the other hand, compared Fig. 9(a) with Fig. 9(b), one can see that influence of laser elevation on self-focusing is stronger than that on spatial coherence (i.e., diffraction).

Fig. 8. Beam width w_tar on the target versus σ₀ and log(C₀), P=1000P_crGs. (a) h₁=0; (b) h₁=6 km.

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Fig. 9. Beam width w_tar on the target versus P/P_crGs and σ₀, C₀=1.7×10⁻¹⁶m^-2/3. (a) h₁=0; (b) h₁=6 km.

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5. Optimization of the optics system and laser

In the section 4, we prove that the turbulence and self-focusing effects can be suppressed by increasing the laser elevation. Furthermore, the influence of laser elevation on turbulence effect is stronger than that on self-focusing effect, and influence of laser elevation on self-focusing effect is stronger than that on diffraction effect. In this section, the optimal focal length and wavelength are proposed to decrease the beam spot size on the target.

5.1 Optimal focal length

The changes of beam width w_tar on the target versus the focal length f are shown in Fig. 10. One can see that there exists a minimum of w_tar when the focal length reaches f_opt. Letting ∂w/∂f=0 in Eq. (19), we can obtain the expression of f_opt, i.e.,

(23)$$\frac{1}{{{f_{\textrm{opt}}}}} = \frac{1}{{z - {h_1}}} - \frac{1}{{{f_\textrm{s}}}}.$$

Substituting from Eq. (5) into Eq. (23), one can see that the optimal focal length f_opt is a function of the beam power P. The changes of the optimal focal length f_opt versus the relative beam power P/P_crGs are shown in Fig. 11. One can see that there are two branches in Fig. 11, and the left and right branches are corresponding to those for the R_s>z-h₁ and R_s<z-h₁ cases respectively. For the R_s>z-h₁ case, a positive lens is needed, and f_opt increases as P increases. However, for the R_s<z-h₁ case, a negative lens is needed to compensate the self-focusing effect, and |f_opt| decreases as P increases.

Fig. 10. Beam width w_tar on the target versus the focal length f, P=500P_crGs, C₀=1.7×10⁻¹⁶ m^-2/3, σ₀=1 m, and the target altitude z=500 km.

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Fig. 11. Optimal focal length f_opt versus the relative beam power P/P_crGs, σ₀=1 m.

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5.2 Optimal wavelength

The changes of beam width w_tar on the target versus the wavelength λ are shown in Fig. 12. One can see that there exists a minimum of w_tar when the wavelength reaches λ_opt. Letting ∂w/∂λ=0 in Eq. (19), we can obtain the expression of λ_opt, i.e.,

(24)$${\lambda _{\textrm{opt}}}\textrm{ = }2.67\pi {\left\{ {5\left( {\frac{1}{{w_0^2}} + \frac{1}{{\sigma_0^2}}} \right){{\left[ {1.46\int_{{h_1}}^z {C_n^2} {{\left( {\textrm{1} - \frac{h}{{z - {h_1}}}} \right)}^{{5 / 3}}}\textrm{d}h} \right]}^{{{\textrm{ - }6} / 5}}}} \right\}^{{{\textrm{ - 5}} / {12}}}}\textrm{.}$$

Equation (24) indicates λ_opt is independent of P, i.e., λ_opt is independent of the self-focusing effect. The physical reason is that the change of beam width due to nonlinear self-focusing is independent of wavelength (see Eq. (19)). It is known that, as the wavelength decreases, the diffraction effect becomes weaker, but the atmospheric turbulence effect becomes stronger. Thus, there exists an optimal wavelength λ_opt to reach the minimum of the beam spot size on the target. The changes of the optimal wavelength λ_opt versus log(C₀) and σ₀ are shown in Figs. 13(a) and 13(b), respectively. One can see that λ_opt increases as log(C₀) and σ₀ increase. In fact, the optimal wavelength λ_opt involves a trade-off between diffraction effect and turbulence effect.

Fig. 12. Beam width on the target versus the wavelength λ, P=1000P_crGs, C₀=1.7×10⁻¹⁶m^-2/3, σ₀=1 m.

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Fig. 13. Optimal wavelength λ_opt versus (a) log(C₀) and (b) σ₀.

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In practice, the transmission window of the atmosphere should be considered when the optimal wavelength is determined. One of the transmission windows of the atmosphere is 3-5μm. The optimal wavelength λ_opt≈2.8μm in Fig. 12. Furthermore, Fig. 12 shows that the beam spot size on the target is also small when the wavelength λ≈3μm. It is noted that the optimal wavelength λ_opt≈2.8μm is obtained only under the calculation parameters adopted in this paper. In fact, the optimal wavelength dependents on the beam propagation path, the atmospheric environment, the initial beam coherence, the initial beam width, etc. (see Eq. (24)).

6. Conclusions

In this paper, the characteristics of high-power partially coherent laser beams propagating upwards in the turbulent atmosphere are studied, where the principal features of diffraction, nonlinear self-focusing and turbulence are considered. Based on the TW model, the analytical formulae of the cross-spectral density function, the intensity, the beam width w and the coherence width σ are derived by using the quadratic approximation of nonlinear phase shift. Furthermore, the validity of the quadratic approximation and the analytical formula derived in this paper are confirmed by using the numerical results. It is shown that the w decreases as the initial spatial coherence width σ₀ increases, and there exists a minimum of σ as the propagation distance z increases. It is found that the atmospheric turbulence plays an important role in beam propagation characteristics. But the turbulence and self-focusing effects can be suppressed by increasing the laser elevation. Furthermore, the influence of laser elevation on turbulence effect is stronger than that on self-focusing effect, and influence of laser elevation on self-focusing effect is stronger than that on diffraction effect. In particular, the optimal focal length and wavelength are proposed to decrease the beam spot size on the target. It is noted that the analytical formulae obtained in this paper are more general, which can reduce to those of fully coherent laser beams when σ₀ → ∞. The results obtained in this paper will be useful for applications, such as space-debris removal and launching small probes into Low Earth Orbit, etc.

Funding

National Natural Science Foundation of China (61775152).

Acknowledgments

The authors are very thankful to the reviewers for their very valuable comments.

Disclosures

The authors declare no conflicts of interest.

References

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Characteristics of high-power partially coherent laser beams propagating upwards in the turbulent atmosphere

Abstract

1. Introduction

2. Theoretical model and analytical formulae

3. Changes of beam propagation characterizations

4. Influence of laser elevation on the self-focusing, turbulence and diffraction effects

5. Optimization of the optics system and laser

5.1 Optimal focal length

5.2 Optimal wavelength

6. Conclusions

Funding

Acknowledgments

Disclosures

References

Cited By

Figures (13)

Equations (24)

Optics Express