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Abstract
The focusability of coherently combined beams (CCBs) propagating from orbit through the turbulent atmosphere to the ground is studied, where the diffraction, self-focusing and turbulence effects are considered. It is shown that the spot size on the ground of CCBs is much smaller than that of incoherently combined beams (ICBs). The analytical expression of the B integral of CCBs propagating in the turbulent atmosphere is derived, and an effective design rule for the CCB power transportation without filamentation is presented. It is found that the focusability of CCBs propagating in the turbulent atmosphere can be improved by the gradient power distribution method, and the spot size on the ground can always be reduced below the diffraction limit. Furthermore, the optimal gradient power distribution to reach the highest focusability on the ground without filamentation is presented.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
It is very important to study the focusability of high-power laser beams propagating from orbit through the turbulent atmosphere to the ground for applications of wireless energy transportation [1]. For example, the accumulated solar energy at orbit stations can be used to produce a laser beam, and the laser beam is utilized for transport of converted solar energy to the ground by laser-based orbit systems [2,3]. One of key problems is that a large focusing optics at orbit stations and a large receiving facility on the ground would be required even for a diffraction limited beam. However, it was shown that the spot size on the ground can be compressed due to the nonlinear self-focusing effect in the atmosphere because the beam power is above the critical power [1]. Furthermore, it is found that the beam filamentation and collapse can be avoided even if the beam power is well above the critical power because the earth’s atmosphere is inhomogeneous [1], which is different from the behavior when a high-power laser beam propagates through the homogeneous atmosphere [4–7]. Our group indicated that the spot size on the ground can be compressed further by using the negative spherical aberration [8], and a partially coherent beam (PCB) has higher threshold critical power than a fully coherent beam (FCB) although a FCB can be more strongly compressed on the ground than a PCB [9]. However, these studies are restricted to the propagation of one single laser beam from orbit through the atmosphere to the ground [1,8,9].
It is known that the output power and the beam quality can be improved by using the beam combination [10,11]. Recently, our group indicated that an incoherently combined beam (ICB) has the advantage to deliver powerful laser beam compared with a single laser beam [12]. On the other hand, the atmospheric turbulence effect will be encountered when a laser beam propagates through the atmosphere, and the beam quality will be reduced due to turbulence [13–17]. However, in these studies [1,8,9,12], the atmospheric turbulence effect has not been involved.
The B integral can characterize quantitatively the beam quality degradation due to self-focusing effect [18]. Until now, the B integral of laser beams under the atmospheric turbulence hasn’t been studied, and the B integral of array beams in the atmosphere also hasn’t been investigated. In vacuum, the beam quality in far field of coherently combined beam (CCB) is better than that of ICB [19,20]. In addition, the beam quality of CCBs propagating in vacuum may be improved by the gradient power distribution method [21]. In this paper, the focusability of CCBs propagating from orbit through the turbulent atmosphere to the ground is studied, where the diffraction, self-focusing and turbulence effects are considered. The main questions investigated in this paper are as follows: Is a CCB is more beneficial to reduce the ground receiver size than an ICB in the turbulent atmosphere? What is the rule for maximal compression without filamentation of transported CCBs in the turbulent atmosphere when the B integral is taken as characteristic parameter of the beam quality? How to further improve the focusability of CCBs in the turbulent atmosphere?
2. Theoretical model
In this paper, the coherent beam combination is assumed, and an array with hexagonal symmetric arrangement is considered (see Fig. 1). In Fig. 1, 2a and D are the diameter of sub-aperture and conformal aperture, respectively, d is the distance between centre of neighbouring sub-aperture, and l is the number of hexagonal rings (l = 0, 1, 2, 3, …) with a relationship of the number N of sub-apertures (i.e., N = 1 + 3(l + l2)). When l = 0, 1, 2, 3, one has N = 1, 7, 19, 37, respectively.
It is known that the sub-aperture filling factor fs and the conformal aperture filling factor fc are two important parameters to describe the performance of array beams. The sub-aperture filling factor is defined as fs = w0/a [22], where w0 is the beam width of elements (Gaussian beams in the sub-aperture). The fs is used to describe the energy loss caused by the sub-aperture, i.e., the more energy will be lost as fs increases. The conformal aperture filling factor is defined as fc = 2a/d [23]. The fc is used to describe the compactness of the array, i.e., a larger value of fc means that the more compact of the array is. For a certain sub-aperture (i.e., a certain value of a), fs and fc depend on w0 and d, respectively, and D = (4 l/fc + 2)a.
In this paper, assume that a CCB propagates vertically from orbit (z = H) to the ground (z = 0) through the turbulent atmosphere, where the diffraction, self-focusing (because the beam power is above the critical power of self-focusing effect) and turbulence effects are considered. At the source plane (z = H), the initial Gaussian field of the jth element with hard-edged aperture of the CCB is written as
whereFor the coherent beam combination, the initial field of CCBs is expressed as
The nonlinear Schrödinger equation can be used to describe the propagation of high-power laser from orbit through the turbulent atmosphere to the ground, i.e. [1,24],
The nonlinear refractive index n2 of the atmosphere is a function of the altitude z, i.e. [1], $n_2(z)=n_2(0) \exp (-z / h)$, where h = 6 km, and n2(0) = 5.6 × 10−19 cm2/W is the refractive index at the ground. The value of n2 decreases as z increases, and the self-focusing effect of the atmosphere can be ignored when the altitude z is high enough (e.g., when z > 40 km, one has n2(z) / n2(0) < 0.0013).
Based on Eq. (5), we design a computer code to simulate numerically the propagation of the CCB in the turbulent atmosphere from orbit through to the ground by using the multi-phase screen approach and the discrete Fourier transform method [1,12]. In the computer code, the phase screen is arranged non-uniformly, and the distance between two phase screens increases with altitude because the atmosphere density decreases with altitude. Assume that the laser beam propagation distance is divided into Q parts. Let E(x, y, zq) be the complete solution to Eq. (5) at zq plane (q = 1, 2, …, Q), and the solution to Eq. (5) at zq + 1 = zq + Δz (Δz is the propagation distance) plane can be written as [24,25]
In this paper, the phase screen of atmospheric turbulence is generated by the power spectrum inversion method, and the van Karman power spectrum is adopted, i.e. [27],
In this paper, the propagation of CCBs from orbit (z = H) to the ground (z = 0) can be divided into two stages, i.e., the first stage propagation: from z = H to 40 km in vacuum (only the diffraction effect is considered), and the second stage propagation: from z = 40 to 0 km in the inhomogeneous turbulent atmosphere (the diffraction, self-focusing and turbulence effects are all considered). The calculation parameters are taken as H = 500 km and λ = 0.8 µm, and the value of the critical power of a Gaussian beam at $z=h=6\,{\rm km}$ (i.e., Pcr = 0.93λ2/2πn0n2 = 4.6 GW) is adopted to normalize the beam power.
In numerical examples of this paper, the result in the atmospheric turbulence is the average result for 1000 realizations because of the randomness of turbulence. Figure 2 shows 3D intensity distributions of CCBs propagating from orbit through the turbulent atmosphere to the ground. The CCBs are compressed because of self-focusing effect (see Figs. 2(a) and 2(b), Figs. 2(d) and 2(e)), but the self-focusing effect is suppressed by the atmospheric turbulence (see Figs. 2(b) and 2(c), Figs. 2(e) and 2(f)). Comparing Fig. 2(c) with Fig. 2(f), one can see that the influence of turbulence on beam spreading becomes weaker as D increases, which is different from the behavior in the homogeneous atmospheric turbulence [29]. The physical reason is that the diffraction effect becomes weaker as D increases, which results in a decrease of the beam size entering the atmosphere, and so the influence of turbulence on the beam spreading becomes weaker.
Fig. 2. 3D intensity distributions on the ground of CCBs, Cn2(0) = 1.7 × 10−14m−2/3, L0 = 10 m. (a)-(c): D = 3 m, P/Pcr = 5.8; (d)-(f): D = 5 m, P/Pcr = 2.
In this paper, the focusability is characterized by the beam width w, which is defined as the radius of the “bucket” corresponding to $PIB = {{\int_0^w {I(r,z)r\textrm{d}r} } / {\int_0^\infty {I(r,z)r\textrm{d}r} }} = 63\%$. Figure 3 shows the beam width w of CCBs on the ground versus L0. One can see that w increases as L0 increases, and w decreases due to self-focusing effect.
Fig. 3. Beam width w of CCBs on the ground versus L0, Cn2(0) = 1.7 × 10−14m−2/3. Solid curve: diffraction & self-focusing & turbulence effects; Dashed curve: diffraction & turbulence effects.
3. Influence of filling factors on the propagation efficiency of CCBs in the atmosphere
In this section, the influence of two kinds of filling factors (fs and fc) on the propagation efficiency of CCBs propagating from orbit through the atmosphere to the ground is studied numerically. The propagation efficiency can denote both the energy loss due to the truncation effect of the hard-edged aperture and the beam power encircled in the bucket (PIB), which is defined as [22]:
In this paper, rb = 0.61λz/D is taken as the radius of bucket at the ground.
For the different values of P, D, N and fc of CCBs propagating from orbit through the atmosphere to the ground, changes of JPIB versus the sub-aperture filling factor fs are shown in Figs. 4(a), 4(b), 4(c) and 4(d), respectively. There exists a maximum of JPIB versus fs, i.e., for the different cases (see Figs. 4(a), 4(b), 4(c) and 4(d)) there always exists an optimal sub-aperture filling factor fsopt = 0.89 in the atmosphere, which is consistent with the result in vacuum [30]. The physical explanations are shown as follows. As fs increases, in the far field the sidelobe intensity decrease and the main lobe intensity increases, but the energy loss also increases. Thus, there will exist an optimal sub-aperture filling factor (fsopt) to maximize the JPIB in the far field because of the trade-off of the two physical mechanism. Furthermore, the self-focusing effect of laser beams in the atmosphere is proportional to intensity. Therefore, fsopt in atmosphere is in agreement with that in vacuum.
On the other hand, as P and D increase (see Figs. 4(a) and 4(b)) or N decreases (see Fig. 4(c)), JPIB increases. Furthermore, JPIB increases as the conformal aperture filling factor fc increases (see Fig. 4(d)). Based on the results obtained in section 3, to improve the focusability on the ground, the values of the two kinds of filling factors are adopted as fs = 0.89 and fc = 1 in numerical simulations in this paper, unless otherwise specified.
4. Comparison of the focusability between CCBs and ICBs in the turbulent atmosphere
In the atmosphere, a high-power laser beam is compressed due to self-focusing effect, but the filamentation will take place when the beam power is over a threshold critical power. In this paper, assume that the threshold critical power reaches when the main lobe is about to be unable to maintain the Gaussian-like shape (i.e., just before beam filamentation). In this section, a comparison of the focusability between CCBs and ICBs propagating from orbit through the turbulent atmosphere to the ground is studied.
Under different conditions, changes of the beam width w on the ground of CCBs and ICBs versus D are shown in Figs. 5(a) and 5(b), respectively, where the maximal beam compression without filamentation is considered. Both for the CCB and the ICB cases, w decreases due to self-focusing effect, w increases because of turbulence effect, and w decreases as D increases.
Fig. 5. Beam width w on the ground versus D, N = 7, Cn2(0) = 1.7 × 10−14m−2/3, L0 = 10 m. (a) CCB; (b) ICB.
In Fig. 5(a), comparing the black curve (i.e., diffraction & turbulence & self-focusing effects) with the green curve (i.e., diffraction effect), one can see that there exists a cross point between the two curves. It means that the CCB can be compressed without filamentation and its spot size on the ground can be reduced below the diffraction limit only when D is large (e.g., D > 3.3 m in Fig. 5(a)).
On the other hand, from Fig. 5(b) (see black and green curves), one can see that for the ICB, its spot size on the ground always can be reduced below the diffraction limit. The physical reason is that the influence of turbulence on the beam spreading for the ICB is weaker than that for the CCB. However, comparing Fig. 5(a) with Fig. 5(b), one can see that the beam width w of CCBs is much smaller than that of ICBs. It means that a CCB is more beneficial to reduce the receiver size than an ICB for array laser power transportation from orbit through the turbulent atmosphere to the ground. Thus, only the CCB is considered in sections 5 and 6.
5. B integral of CCBs in the turbulent atmosphere
The B integral can quantitatively describe the nonlinear phase modulation caused by the self-focusing effect [18]. The value of the B integral usually should not exceed several units to avoid filamentation [6]. In this section, the expression of the B integral of CCBs propagating in the turbulent atmosphere are derived, the influence of N, D and Cn2(0) on the B integral is studied, and an effective design rule for the CCB power transportation in the turbulent atmosphere is presented.
5.1. Analytical expression of the B integral in the turbulent atmosphere
The B integral of laser beams propagating through the turbulent atmosphere can be expressed as
At the source plane (z = H), the initial cross-spectral density function of CCBs is written as
Based on the generalized Huygens-Fresnel principle and the quadratic approximation of the Rytov’s phase structure function, the intensity of CCBs propagating in atmospheric turbulence is expressed as [27,32]
Substituting Eqs. (11) and (12) into Eq. (13), and applying the integral formula $\int_{ - \infty }^{ + \infty } \exp ( - \alpha {x^2} + \beta x) \textrm{d}x = \sqrt {{\mathrm{\pi } / \alpha }} \exp ({{\beta ^2}} / {4\alpha })$, after tedious integral calculations we obtain the intensity of CCBs propagating downwards through the turbulent atmosphere, i.e.,
Letting x = y = z = 0 in Eq. (14), we obtain the analytical expression of the maximum intensity I0 at the ground. Substituting this expression of I0 into Eq. (10), we can obtain the analytical expression of the B integral of CCBs propagating from orbit through the turbulent atmosphere to the ground.
The changes of the B integral versus N and D are shown in Figs. 6 and 7, respectively. Figure 6 indicates that the B integral decreases due to turbulence effect. The physical reason is that the self-focusing effect is suppressed by the atmospheric turbulence. Furthermore, the B integral decreases slightly as N increases. Figure 7 shows that the B integral increases as D increases because the self-focusing effect becomes stronger. In addition, the B integral almost remain unchanged when N > 7, which is in agreement with that obtained in Fig. 6.
Fig. 6. With or without turbulence effect, the B integral on the ground versus N, P/Pcr = 2, D = 5 m, Cn2(0) = 1.7 × 10−14m−2/3.
Fig. 7. For different values of N, the B integral on the ground versus D, P/Pcr = 2, Cn2(0) = 1.7 × 10−14m−2/3.
5.2. Fitting expression of the maximum B integral in the turbulent atmosphere
In this section, the expression of the maximum B integral of CCBs propagating from orbit through the turbulent atmosphere to the ground is obtained by fitting method. For different values of beam parameters (e.g., D, N) and turbulence parameter (e.g., Cn2(0)), the threshold critical power is calculated by numerical simulation method. And then, substituting values of the threshold critical power into Eq. (14) together with Eq. (10), the values of the maximum B integral can be obtained (see the black dots in Fig. 8).
Fig. 8. For different N, Bmax on the ground versus D and Cn2(0), L0 = 10 m. Black dots: numerical simulation results; Curve surface: fitting surface by using Eq. (15).
For different N, changes of the maximum B integral (Bmax) versus D and Cn2(0) are shown in Fig. 8, where the black dots denote the numerical simulation results. Based on the black dots in Fig. 8, the fitting expression of Bmax of CCBs propagating from orbit through the turbulent atmosphere to the ground is obtained, i.e.,
Table 1. Values of the coefficients in Eq. (15).
Figure 8 indicates that for different values of N, Bmax decreases as D increases, and Bmax increases slightly as log Cn2(0) increases. Furthermore, Bmax decreases slightly as N increases (see Figs. 8(a) and 8(b)), and Bmax almost remain unchanged when N > 7 (see Figs. 8(b) and 8(c)). These results are in agreement with those of the B integral (see Figs. 6 and 7).
When B < Bmax, a CCB can be compressed as a whole due to the self-focusing effect without filamentation. Thus, we present an effective design rule for the array laser power transportation from orbit through the turbulent atmosphere to the ground.
6. Improvement of the focusability of CCBs in the turbulent atmosphere
In this section, we demonstrate that the focusability of CCBs propagating from orbit through the turbulent atmosphere to the ground can be improved further by the gradient power distribution method, and present the optimal gradient power distribution to reach the highest focusability on the ground (i.e., the spot size on the ground is smallest). Note that the results are obtained when the total power of CCBs with the gradient power distribution is the same as that with the equal power distribution for the maximal beam compression without filamentation.
The power gradient of CCBs is defined as the ratio Pj(0): Pj(l), where Pj(0) is beam power of the central element, and Pj(l) is beam power of the element on the l ring. By using numerical simulation method, we obtain the optimal gradient power distribution for different values of N, which are shown as follows: Pj(0): Pj(1) = 1: 0.5 when N = 7, Pj(0): Pj(1): Pj(2) = 1: 0.8: 0.4 when N = 19, Pj(0): Pj(1): Pj(2): Pj(3) = 1: 0.8: 0.6: 0.3 when N = 37, Pj(0): Pj(1): Pj(2): Pj(3): Pj(4) = 1: 0.9: 0.75: 0.5: 0.3 when N = 61, Pj(0): Pj(1): Pj(2): Pj(3): Pj(4): Pj(5) = 1: 0.95: 0.9: 0.7: 0.5: 0.3 when N = 91, and the optimal power gradient distributions (i.e., optimal normalized intensity distributions at the source plane) are shown in Fig. 9.
Both in the turbulent atmosphere and in vacuum, changes of the beam width w on the ground of CCBs with the gradient and equal power distributions versus D are shown Fig. 10. As compared with beam propagation in vacuum, in the turbulent atmosphere the ground spot size decreases further for the gradient power distribution versus that for the equal power distribution. In particular, the spot size on the ground always can be reduced below the diffraction limit because in the far field (i.e., within the atmosphere) the beam power in the main lobe increases due to the gradient power distribution of CCBs.
Fig. 10. For the gradient and equal power distributions, beam width w on the ground versus D, N = 7, Cn2(0) = 1.7 × 10−14m−2/3, L0 = 10 m. Solid line: in the turbulent atmosphere; Dashed line: in vacuum.
The relative variation of beam width is defined as η = (wfree − w) / wfree, where w and wfree are the beam width on the ground of a CCB in the atmosphere and vacuum, respectively. A larger value of η means that the spot size on the ground is more compressed when a CCB propagates in turbulent atmosphere. For the gradient and equal power distributions, changes of η versus N are shown in Fig. 11. One can see that η of CCBs with gradient power distribution is larger than that with equal power distribution. As N increases, the difference of η between the gradient power distribution and the equal power distribution increases (see Fig. 11), because initial intensity envelope is closer to Gaussian-shaped distribution (see Fig. 9). Therefore, the gradient power method produces a smaller spot than the equal power method when a CCB propagates from orbit through the turbulent atmosphere to the ground.
Fig. 11. For the gradient and equal power distributions, changes of η versus N, D = 5 m, P/Pcr = 2, L0 = 10 m, Cn2(0) = 1.7 × 10−14m−2/3.
7. Conclusions
In this paper, the focusability of CCBs propagating from orbit through the turbulent atmosphere to the ground is studied analytically and numerically, where the diffraction, self-focusing and turbulence effects are considered. Firstly, it is shown that the spot size on the ground of CCBs is much smaller than that of ICBs, namely, a CCB is more beneficial to reduce the ground receiver size than an ICB. Secondly, the analytical expression of the B integral and the fitting expression of the maximum B integral of CCBs propagating in the turbulent atmosphere are derived, and an effective design rule for the CCB power transportation from orbit through the turbulent atmosphere to the ground without filamentation is presented. It is shown that the B integral decreases (i.e., the self-focusing effect becomes weaker) due to turbulence effect. Finally, it is found that the spot size on the ground of CCBs decreases further for the gradient power distribution versus that for the equal power distribution due self-focusing effect in the turbulent atmosphere, and this advantage increases as the number of sub-apertures increases. Furthermore, it is demonstrated that the spot size on the ground of CCBs with the gradient power distribution can always be reduced below the diffraction limit. In particular, the optimal gradient power distribution to reach the highest focusability on the ground without beam filamentation is presented. The results obtained in this paper are useful for the CCB power transportation from orbit through the turbulent atmosphere to the ground.
Funding
National Natural Science Foundation of China (62375191).
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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