Effect of spatial coherence on laser beam self-focusing from orbit to the ground in the atmosphere

Hanling Deng; Xiaoling Ji; Xiaoqing Li; Hao Zhang; Xianqu Wang; Yuqiu Zhang

doi:10.1364/OE.24.014429

1. Introduction

Energy and environment issues are two serious problems humanity will face in the future. Harnessing and accumulation of solar energy at orbit stations for further wireless laser power transportation to Earth is one of the global concepts of renewable energy sources [1,2]. Recently, the emergence of a multi-channel fiber-based Coherent-Amplifying-Network or CAN laser potentially enables such capability for space based missions [3]. If this technology is applied in practice, one of the main problems is large precise orbital optics and large ground receivers would be required. In 2009, Rubenchik et al. proposed to exploit a self-focusing effect in the atmosphere to assist delivering powerful laser beams from orbit to the ground, which can greatly relax the requirements for the orbital optics and ground receivers [4].

It is known that a steady-state channel of powerful laser beams is formed when the self-focusing is compensated by diffraction [5–8]. When the beam power is over a threshold critical power, the steady-state propagation is unstable in homogeneous atmosphere because filamentation, collapse, and uncontrolled ionization result in breaking of the beams [8–10]. However, this situation can be different for propagation from orbit through Earth’s inhomogeneous atmosphere. Rubenchik et al. demonstrated that when the self-focusing length is comparable with the atmosphere height the whole beam can be compressed smoothly [4]. For high-power regimes, beam aberrations are likely to be much more important than in low-power situations. In 2015, we showed that for the small orbital mirror size, a Gaussian beam with negative spherical aberration might be more strongly compressed without beam splitting than that without spherical aberration, and its spot size on the ground can be reduced well below the diffraction limit [11]. In addition, other space applications could benefit from the possibility of controlling from orbit the generation of laser filaments in the atmosphere. For instance, I. Dicaire et al. recently proposed a theoretical proof-of-concept of laser filamentation and remote supercontinuum generation from orbital altitudes, which might provide the basis for a new remote sensing tool for atmospheric research: an Earth-orbiting white-light lidar [12].

In practice, a partially (spatial) coherent beam (PCB) is often encountered. Atmospheric turbulence results in spatial coherence degradation [13], and changes of spatial coherence in atmospheric turbulence were studied [14, 15]. Furthermore, it was demonstrated that a PCB is less sensitive to the effect of turbulence than a fully coherent one [16, 17], and, as a consequence, the turbulence-induced spatial broadening of the fully coherent beam is a limiting factor in most applications. Until now, the effect of spatial coherence on laser beam self-focusing hasn’t been investigated in inhomogeneous atmosphere. In this paper, by using the numerical simulation method, we study the sensitivity of a PCB to the effect of self-focusing, and the relation between the spatial coherence and the threshold critical power. In addition, we investigate the rule for maximal compression without beam splitting of the transported PCB from orbit to the ground.

2.Theoretical model

2.1 Linear theory

Suppose z is the direction of beam propagation from the orbit to the ground, with z = 0 the sea level. In this paper, the Gaussian Schell-mode (GSM) beam is adopted as a typical example of PCBs. The source cross-spectral density function of a GSM beam with partial spatial coherence at the height of the orbit, z = F, can be expressed as [18]

W (z = F, \vec{r_{1}}, \vec{r_{2}}) = \sqrt{I (\vec{r_{1}}) I (\vec{r_{2}})} μ (\vec{r_{1}} - \vec{r_{2}}),

where F is the focal distance or the orbit height, the averaged intensity

I (\vec{r})

and the spectral degree of source coherence

μ (\vec{r_{1}} - \vec{r_{2}})

can be written as

I (\vec{r}) = I_{0} \exp (- 2 r^{2} / w_{0}^{2})

and

μ (\vec{r_{1}} - \vec{r_{2}}) = \exp [- {(\vec{r_{1}} - \vec{r_{2}})}^{2} / 2 σ_{0}^{2}]

respectively, with I₀ being a constant, w₀ being the waist width, and σ₀ being the lateral correlation radius which determines the spectral degree of source coherence. In this paper, τ = σ₀/w₀ is taken as the correlation parameter to represent the spatial coherence. The larger the value of τ is, the better the source coherence is. When τ → ∞, it is corresponding to a fully coherent beam (Gaussian beam).

In the linear theory and the common case of long distance focusing with a mirror, the location of the minimum (waist) for a PCB without truncation is at

F^{'} = \frac{F}{1 + {(R_{\min} / R_{0})}^{2}} = \frac{F}{1 + {[λ F \sqrt{1 + 1 / τ^{2}} / (π D R_{0})]}^{2}},

and the radius of the focal spot is

{R^{'}}_{\min} = \frac{R_{\min}}{\sqrt{1 + {(R_{\min} / R_{0})}^{2}}} = \frac{λ F \sqrt{1 + 1 / τ^{2}} / (π D)}{\sqrt{1 + {[λ F \sqrt{1 + 1 / τ^{2}} / (π D R_{0})]}^{2}}},

where

R_{\min} = λ F \sqrt{1 + 1 / τ^{2}} / (π D)

,

D = 2 R_{0}

is the diameter of the mirror, and

R_{0} = w_{0} / \sqrt{2}

is called the mirror radius. When R₀ >> R_min, Eqs. (2) and (3) reduce to

F' = F

and

{R^{'}}_{\min} = R_{\min}

. It means that when the focal spot is much smaller than the focusing mirror, the PCB is focused on the ground with the radius

R_{ground} = R_{\min} = λ F \sqrt{1 + 1 / τ^{2}} / (π D)

. It shows that R_ground decreases as τ increases. In particular, for a fully coherent beam (i.e., τ → ∞), the beam footprint on the ground is smallest (i.e., R_ground = R_min = λF/πD). It can be shown that for a PCB in the linear theory (2) and (3) under conditions of maximal compression on the ground, the radius of the mirror R₀ cannot be less than a certain minimal value given by

R_{0} \geq R_{0}_{, \min} = \sqrt{λ F \sqrt{1 + 1 / τ^{2}} / (2 π)}

. In physical terms, this means that at small mirror size the PCB cannot be focused at all, because F in this case becomes comparable with the Rayleigh length. Furthermore, a linear compression spot of the PCB with such a “minimal-radius orbital mirror” is limited by

{R^{'}}_{\min}^{(1)} = R_{0, \min} / \sqrt{2}

. It can be seen that the minimal values

R_{0, \min}

and

{R^{'}}_{\min}^{(1)}

decrease as τ increases.

2.2 Nonlinear theory

It is known that the nonlinear Schrödinger equation can describe the principal features of diffraction and Kerr nonlinearity of laser beams propagating from orbit to the ground, which can be expressed as [4]

2 i k \frac{\partial A}{\partial z} + \nabla_{⊥}^{2} A + 2 k^{2} \frac{n_{2}}{n_{0}} {| A |}^{2} A = 0,

where A is the envelope of the electric field under the standard paraxial approximation, k is the wave-number related to the wave length λ by

k = 2 π n_{0} / λ

,

n_{0}

is the linear refractive index,

n_{2}

is the nonlinear refractive index,

\nabla_{⊥}^{2}

is the transverse Laplace operator which describes transverse diffraction of laser beams. In numerical simulation the multi-phase screen approach can be applied to solve Eq. (4) [19].

The nonlinearity in Eq. (4) is a function of altitude. The nonlinearity refractive index can be obtained by interpolating as exponential in the isothermal atmosphere, which is expressed as $n_{2} (z) = n_{2} (0) \exp [- (z / h)]$ , where h = 6km, and n₂(0) = 5.6 × 10⁻¹⁹cm²/W is the refractive index on the ground [4]. It is noted that self-focusing in homogeneous media starts when the beam power exceeds the critical value $P_{c r} = 0.93 λ_{0}^{2} / (2 π n_{0} n_{2})$ [20, 21]. In this paper, λ₀ = 0.8μm is adopted, and then we have P_cr = 4.6GW at z = h = 6km.

The GSM beam is an analytically tractable model. However, a practical PCB may not follow the GSM theory and is better examined in detail through some type of numerical simulation approach such as a wave optics simulation. Xiao and Voelz proposed that in numerical simulation the partial coherence may be modeled by a random phase screen which changes randomly with a Gaussian spatial coherence function [22], and this random phase screen approach was applied to study the propagation of pseudo-partially GSM beams through turbulence [23].

According to the random phase screen approach [22], the source field of a PCB at the height of the orbit, z = F, can be expressed as

A (z = F, r; t) = \sqrt{\frac{P}{π R_{0}^{2}}} \exp [- \frac{(1 + i C_{0})}{2 R_{0}^{2}} r^{2}] \exp [i ξ (r; t)],

where

C_{0} = k R_{0}^{2} / F

, P is the initial beam power at z = F, and

ξ (r; t)

is the random phase function.

ξ (r; t)

is modeled as

ξ (r; t) = v (r; t) \otimes f (r)

, where

\otimes

denotes a spatial convolution,

v (r; t)

is a spatially uncorrelated random signal with standard deviation σ_v which is related to the amplitude variation of the screen, and f(x, y) is a Gaussian function with standard deviation σ_f which is a transverse spatial correlation length parameter, i.e [22],

f (r) = \frac{1}{2 π σ_{f}^{2}} \exp [- r^{2} / (2 σ_{f}^{2})]

If $σ_{v}^{2} / (4 π σ_{f}^{2}) > > 1$ , the following equation is satisfied, i.e [22],

σ_{0}^{2} = \frac{8 π σ_{f}^{4}}{σ_{v}^{2}} .

Based on the multi-phase screen approach and the random phase screen approach [22], we design a computer code of propagation of a PCB (i.e., pseudo-partially GSM beam) through the atmosphere to calculate the intensity and the spot size on the ground. In this paper, we consider this model, i.e., energy transport from a low earth orbit in which a PCB is focused by an orbital mirror and propagates to the ground from height of 500km, and the variation of atmospheric refractive index starts to affect the beam propagation below approximately 20km.

3. Numerical simulation results and analysis

Figure 1 shows the intensity distribution on the ground for PCBs with different values of the correlation parameter τ. Compared with the linear propagation in free space (see Fig. 1(a)), in the atmosphere PCBs are compressed as a whole even if the beam power P is much larger than the critical value P_cr (i.e., P/P_cr = 5) due to self-focusing effect (see Fig. 1(b)). Furthermore, from Fig. 1(b) it can be seen that a fully coherent beam (i.e., Gaussian beam with τ → ∞) is more strongly compressed than a PCB, and has the leptokurtic beam profile. It is noted that in this paper higher order effects (e.g., plasma defocusing, multiphoton absorption, etc.) are neglected, but in Ref [12]. the higher initial beam power is adopted and the higher order effects are considered.

Fig. 1 Intensity distribution on the ground for PCBs with different values of the correlation parameter τ. R₀ = 0.7m, C₀ = 7.697, P/P_cr = 5.

Download Full Size | PDF

In this paper, assume that the threshold critical power reaches when the ring structures take place, and the beam width w is defined as the diameter at half maximum intensity. Figure 2 shows changes of the compressed beam width w on the ground versus the relative power P/P_cr for PCBs with different values of the correlation parameter τ. It can be seen that w decreases as P/P_cr or R₀ increases, and the threshold critical power increases as R₀ decreases. Furthermore, the compressed beam width w on the ground and the threshold critical power are dependent on the correlation parameter τ, i.e., the w and the threshold critical power increase when τ decreases. Namely, the compressed beam width w on the ground for PCBs is larger than that of fully coherent beams, but PCBs can be compressed as a whole even if the initial power is much higher.

Fig. 2 Changes of the compressed beam width w on the ground versus the relative power P/P_cr for PCBs with different values of the correlation parameter τ. (a) R₀ = 0.7m, C₀ = 7.697; (b) R₀ = 0.3626m, C₀ = 1.291.

Download Full Size | PDF

The three-dimensional (3D) intensity distributions on the ground for PCBs with different correlation parameter τ are shown in Fig. 3. It can be seen clearly that the larger the value of τ is, the more strongly the PCB is compressed. Namely, the beam width on the ground decreases and the maximum intensity on the ground increases as correlation parameter τ increases.

Fig. 3 3D intensity distributions on the ground for PCBs with different correlation parameter τ. (a)-(d) R₀ = 0.7m, C₀ = 7.697, P/P_cr = 4; (e)-(h) R₀ = 0.3626m, C₀ = 1.291, P/P_cr = 80.

Download Full Size | PDF

Evolution of the beam width w on propagation for PCBs with different correlation parameter τ both in the atmosphere and in free space is shown in Fig. 4, where Fig. 4(b) is the evolution of the beam width w on propagation just below 20km. It is seen from Fig. 4(b) that the curve in the atmosphere separate from the curve in free space when PCBs enter the atmosphere. The larger the τ is, the more sensitive the PCB is to self-focusing effect, and the earlier the separation is. Furthermore, as τ increases, the PCB is more strongly compressed on the ground due to self-focusing effect in comparison with the linear propagation in free space. It is noted that the compressed spot size on the ground of PCBs (e.g., τ = 1.6 in the atmosphere) may be reduced well below the diffraction limit (i.e., τ → ∞ in free space).

Fig. 4 Evolution of the beam width w on propagation for PCBs with different correlation parameter τ. R₀ = 0.7m, C₀ = 7.697, P/P_cr = 5.

Download Full Size | PDF

Figure 5 shows changes of the beam width w on the ground versus the correlation parameter τ. It can be seen that both in free space and in the atmosphere w decreases as τ increases, but the difference of w between in free space and in the atmosphere increases as τ increases. Figure 6 shows changes of the average radial compression ratio $w_{atm} / w_{free}$ on the ground versus the correlation parameter τ, where $w_{atm}$ and $w_{free}$ are the beam width in the atmosphere and free space respectively. It is known that the smaller value of $w_{atm} / w_{free}$ means the spot size on the ground is more strongly compressed due to self-focusing effect. From Fig. 6 it can be seen that $w_{atm} / w_{free}$ decreases as τ increases. It implies that PCBs with better coherence are compressed more strongly on the ground than those with worse coherence.

Fig. 5 Changes of the beam width w on the ground versus the correlation parameter τ. P/P_cr = 5.

Download Full Size | PDF

Fig. 6 Changes of the average radial compression ratio w_atm/w_free on the ground versus the correlation parameter τ. (a) P/P_cr = 5; (b) P/P_cr = 45.

Download Full Size | PDF

Changes of the beam width w and the average radial compression ratio $w_{atm} / w_{free}$ on the ground versus the mirror size R₀ are shown in Figs. 7(a) and 7(b), respectively. It is seen that both in free space and in the atmosphere w decreases as R₀ increases, but the difference of w between in free space and in the atmosphere increases as R₀ increases (see Fig. 7(a)), and $w_{atm} / w_{free}$ decreases as R₀ increases (see Fig. 7(b)). It means for PCBs it is also truth just as that for fully coherent beams, i.e., PCBs can be more strongly compressed on the ground as the mirror size R₀ increases.

Fig. 7 (a) Changes of the beam width w, and (b) the average radial compression ratio w_atm/w_free on the ground versus the mirror size R₀. Empty shapes: P/P_cr = 5; filled shapes: P/P_cr = 45.

Download Full Size | PDF

4. Rule for maximal compression without beam splitting

When P is above the critical value, the beam collapses to a singularity at a distance L (self-focusing length), which for unfocused PCBs is given by $L ~ \frac{π a^{2} (1 + 1 / τ^{2})}{λ \sqrt{P / P_{c r} - 1}}$ , where a represents the Gaussian beam size entering the atmosphere, and a is close to its footprint on the ground when the Rayleigh length is comparable to or longer than that atmosphere height. On condition L ≈h, we can derive the following relation

(\frac{P}{P_{c r}} - 1) D^{4} ~ \frac{4 λ^{2} F^{4} {(1 + 1 / τ^{2})}^{2}}{π^{2} h^{2}} .

Equation (7) presents an effective design rule for maximal compression without beam splitting of transported PCBs. Equation (7) is more general, which can reduce to the result given by Rubenchik et al. (see formula (5) in [4]) when τ → ∞ (i.e., fully coherent beams).

Equation (7) can be confirmed based on the numerical simulation result. Figure 8 shows the result of the numerical modeling, and it indicates that the ratio of D⁴(P/P_cr-1) between a PCB and a fully coherent beam is (1 + 1/τ²)², which confirms that Eq. (7) holds and can be used in an estimate of the PCB or system parameters that allow for laser power transportation from orbit to the ground without beam splitting.

Fig. 8 Dependence D⁴(P/P_cr-1) on power P/P_cr required without beam splitting for PCBs with different correlation parameter τ.

Download Full Size | PDF

5. Conclusion

In this paper, we study the effect of spatial coherence on laser beam self-focusing in the atmosphere to assist delivering powerful laser beams from orbit to the ground. It is found that a fully coherent beam is more strongly compressed on the ground than a PCB, even so, for a PCB the compressed spot size on the ground may be reduced below the diffraction limit due to self-focusing effect, and a PCB has higher threshold critical power than a fully coherent beam. A PCB with better coherence is compressed more strongly on the ground than that with worse coherence, but a PCB with worse coherence has higher threshold critical power than that with better coherence. On the other hand, an effective design rule for maximal compression without beam splitting of the transported PCB presented, which can be used in an estimate of the PCB or system parameters that allow for laser power transportation from orbit to the ground.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (NSFC) under grants 61178070, 61475105. The authors are very thankful to the reviewers for their very valuable comments.

References and links

1. C. E. Davis, “Solar energy system with relay satellite,” U.S. patent 43055550 (December 15, 1981).

2. J. C. Mankins, “Space solar power: a major new energy option?” J. Aerosp. Eng. 14(2), 38–45 (2001). [CrossRef]

3. M. N. Quinn, V. Jukna, T. Ebisuzaki, I. Dicaire, R. Soulard, L. Summerer, A. Couairon, and G. Mourou, “Space-based application of the CAN laser to LIDAR and orbital debris remediation,” EPJ ST 224(13), 2645–2655 (2015).

4. A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, “Laser beam self-focusing in the atmosphere,” Phys. Rev. Lett. 102(23), 233902 (2009). [CrossRef] [PubMed]

5. R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13(15), 479–482 (1964). [CrossRef]

6. V. I. Talanov, “Self-focusing of electromagnetic waves in non-linear media,” Izv. Vyssh. Uchebn, Zaved. Radiofiz. 7, 564 (1964).

7. V. I. Talanov, “Self-modeling wave beams in a nonlinear dielectric,” Sov. Radiophys. 9(2), 260–261 (1967). [CrossRef]

8. R. I. Miller and T. G. Roberts, “Laser self-focusing in the atmosphere,” Appl. Opt. 26(21), 4570–4575 (1987). [CrossRef] [PubMed]

9. V. I. Talanov, “Self-focusing of wave beams in nonlinear media,” JETP Lett. 2(5), 138 (1965).

10. P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15(26), 1005–1008 (1965). [CrossRef]

11. H. Deng, X. Ji, X. Li, and X. Wang, “Effect of spherical aberration on laser beam self-focusing in the atmosphere,” Opt. Lett. 40(16), 3881–3884 (2015). [CrossRef] [PubMed]

12. I. Dicaire, V. Jukna, C. Praz, C. Milian, L. Summerer, and A. Couairon, “Space-borne laser filamentation for atmospheric remote sensing,” Laser Photonics Rev. 10(3), 481–493 (2016). [CrossRef]

13. J. Wu and A. D. Boardman, “Coherence length of a Gaussian-Schell beam and atmospheric turbulence,” J. Mod. Opt. 38(7), 1355–1363 (1991). [CrossRef]

14. X. Ji, X. Chen, S. Chen, X. Li, and B. Lü, “Influence of atmospheric turbulence on the spatial correlation properties of partially coherent flat-topped beams,” J. Opt. Soc. Am. A 24(11), 3554–3563 (2007). [CrossRef] [PubMed]

15. Y. Baykal, “Formulation of correlations for general-type beams in atmospheric turbulence,” J. Opt. Soc. Am. A 23(4), 889–893 (2006). [CrossRef] [PubMed]

16. G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19(8), 1592–1598 (2002). [CrossRef] [PubMed]

17. O. Korotkova and E. Wolf, “Beam criterion for atmospheric propagation,” Opt. Lett. 32(15), 2137–2139 (2007). [CrossRef] [PubMed]

18. E. Tervonen, A. T. Friberg, and J. Turunen, “Gaussian Schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A 9(5), 796–803 (1992). [CrossRef]

19. M. Delfour, M. Fortin, and G. Payr, “Finite-difference solutions of a non-linear Schrödinger equation,” J. Comput. Phys. 44(2), 277–288 (1981). [CrossRef]

20. V. E. Zakharov, “Instability of self-focusing of light,” Sov. Phys. JETP 26(5), 994 (1968).

21. V. E. Zakharov and A. M. Rubenchik, “Instability of waveguides and solitons in nonlinear media,” Sov. Phys. JETP 38, 494 (1974).

22. X. Xiao and D. Voelz, “Wave optics simulation approach for partial spatially coherent beams,” Opt. Express 14(16), 6986–6992 (2006). [CrossRef] [PubMed]

23. X. Qian, W. Zhu, and R. Rao, “Numerical investigation on propagation effects of pseudo-partially coherent Gaussian Schell-model beams in atmospheric turbulence,” Opt. Express 17(5), 3782–3791 (2009). [CrossRef] [PubMed]

Effect of spatial coherence on laser beam self-focusing from orbit to the ground in the atmosphere

Abstract

1. Introduction

2.Theoretical model

2.1 Linear theory

2.2 Nonlinear theory

3. Numerical simulation results and analysis

4. Rule for maximal compression without beam splitting

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (8)

Equations (8)

Optics Express