Self-focusing and group-velocity dispersion of pulsed laser beams in the inhomogeneous atmosphere

Yuqiu Zhang; Xiaoling Ji; Hao Zhang; Xiaoqing Li; Tao Wang; Huan Wang; Yu Deng

doi:10.1364/OE.26.014617

1. Introduction

A large number of space debris poses great threat to human space activities. The pulsed-laser debris removal is an effective method. Phipps et al. gave an analytical model for calculating the coupling coefficient of pulsed-laser debris removal, and discussed the region of vapor-plasma transition related to the optimal coupling [1,2]. Schall et al. discussed the feasibility and basic principle for cleaning space debris by laser radiation [3]. The debris removal facilitated by a ground-based pulsed laser was analyzed by the ‘Orion’ project [4,5]. The National Ignition Facility (NIF) laser system at Lawrence Livermore National Laboratory is designed for inertial-confinement-fusion applications. In 2012, this facility performed target experiments with 1.9 MJ at peak powers of 410 TW [6,7]. Rubenchik et al. proposed that the NIF laser system is a near-perfect fit for orbital-debris-removal applications [8]. However, the problem of powerful laser beam propagation through the atmosphere will be encountered for space-debris removal facilitated by a ground-based pulsed laser. It is important to study the self-focusing effect of a powerful laser beam propagating through the atmosphere for applications [9]. We investigated the effect of spherical aberration on laser beam self-focusing in the atmosphere [10]. Recently, Rubenchik et al. demonstrated that the laser power for the space-debris removal is well above the critical power of self-focusing in the atmosphere, and showed that the self-focusing can noticeably decrease the continuous (CW) laser intensity on the debris target [8]. It is known that CW lasers cannot reach the required intensity on debris target unless an unacceptably large mirror is used [5], and shorter pulses can spend less money [11].

The group velocity dispersion (GVD) effect occurs in propagation for pulsed laser beams. A simple estimate shows that the GVD effect plays almost no role when nanosecond and even picosecond pulses propagate in the atmosphere (e.g., for 2 ps pulse for typical dispersion parameter 0.02 ps²/km of the atmosphere predicts dispersion length exceeding 100 km [12]). However, in this paper we demonstrate that the situation can be different when a powerful laser pulse propagates from the ground to the orbit even if the atmosphere height is only 40 km. We show that changes of the pulse duration and the beam spot size on propagation from the ground to the orbit are noticeable due to the interplay of the GVD effect and the self-focusing effect. On the other hand, in this paper we derive crucial formulae of the modified focal length and the M²-factor and propose the useful methods for space-debris removal facilitated by a ground-based pulsed laser.

2. Theoretical model

For simplicity, we consider a pulsed laser beam propagation along the vertical direction z from the ground to the orbit. The propagation of the pulsed laser beam through the atmosphere can be described by the nonlinear Schrödinger equation [13], i.e.,

\frac{\partial A}{\partial z} = \frac{i}{2 k} \nabla_{⊥}^{2} A - i \frac{β''}{2} \frac{\partial^{2} A}{\partial t^{2}} + i k \frac{n_{2}}{n_{0}} {| A |}^{2} A,

where the terms on the right side of Eq. (1) describe the transverse diffraction, the GVD and the Kerr nonlinearity in turn.

\nabla_{⊥}^{2} = \partial^{2} / \partial x^{2} + \partial^{2} / \partial y^{2}

, A is the field of the pulsed laser beam, k is the wave-number related to the wave length

λ

by

k = 2 π / λ

, n₀ is the linear refractive index, and n₂ is the nonlinear refractive index. In the atmosphere, n₂ is a function of altitude, and can be expressed as

n_{2} (z) = n_{2}_{0} e x p [- z / h]

[8], where h = 6 km and

n_{2}_{0} = 5.6 \times 10^{- 19} {cm}^{2} / W

is the refractive index on the ground.

β'' = \partial {}^{2}k / \partial ω {}^{2}|_{ω {}_{0}}

is the group-velocity dispersion parameter, ω is the angular frequency and ω₀ is the initial angular frequency. In this paper, the normal GVD is considered in the atmosphere (i.e.,

β''

is positive), and

β''

is related to altitude which can be expressed as [14]

β'' = \frac{3 .5 n_{0} (ω) P (z)}{2 π c^{2} λ θ (z)} 10^{15},

where n₀(ω) is the linear refractive index related to the angular frequency ω, θ(z) and P(z) are the temperature and the pressure related to the altitude z (z is in meters). The expressions of θ(z), P(z) and n₀(ω) are shown in [14] (see Eqs. (1), (2) and (4) in [14]). The values of n₂ and

β''

are very small when the altitude from the ground is large enough, and so the self-focusing (Kerr nonlinearity) and GVD are considered just below 40 km in this paper.

The initial field of a pulsed laser beam on the ground is expressed as [15]

A (r, t, z = 0) = \sqrt{\frac{2 P}{π w_{0}^{2}}} \exp [- \frac{r^{2}}{w_{0}^{2}} (1 + \frac{i k w_{0}^{2}}{2 F}) - \frac{t^{2}}{T_{0}^{2}}],

where P is the initial peak power, w₀ is the initial beam width, T₀ is the initial pulse width, and F is the focal distance or the height of orbit.

Based on the split-step and Crank-Nicholson methods [16,17], Eq. (1) can be solved numerically. The required laser-pulse energy for optimal coupling of space-debris removal depends on the beam spot size and the pulse duration on the debris target [5,8]. In this paper, the intensity distribution, the beam width and the pulse width on the debris target are examined by using the computer code designed by us. The mean-squared beam width w and the root-mean-square (RMS) pulse width T are adopted, which are defined as [18,19]

w^{2} = 2 \iint r^{2} {| A (r, z) |}^{2} d^{2} r / \iint {| A (r, z) |}^{2} d^{2} r,

and

T^{2} = 〈 σ^{2} 〉 - {〈 σ 〉}^{2},

respectively, where

〈 σ^{n} 〉 = {\int_{- \infty}^{\infty} t^{n} | A (t, z) |}^{2} d t / {\int_{- \infty}^{\infty} | A (t, z) |}^{2} d t

, and n = 1, 2. Unless specified, the numerical calculation parameters are taken as

w_{0} = \sqrt{2} m

,

λ = 1 .06 μm

, and the orbit height L = 1000 km where the debris is most concentrated [5].

3. Temporal and spatial intensity distributions

The normalized intensity distributions I/I₀ on the debris target for different values of initial pulse width T₀ and peak power P are shown in Fig. 1, where I₀ is the linear peak intensity on the target, P_cr is the critical power in a homogeneous medium (i.e., P_cr = λ²/2πn₀n₂₀ = 4.3GW [8]). One can see that the temporal pulse splitting and the spatial side lobe appear on the debris target for nanosecond and picosecond pulses when P is high enough (see Figs. 1(a) and 1(b)), but these phenomena don’t appear for the femtosecond pulse (see Fig. 1(c)).

Fig. 1 Normalized intensity distributions I/I₀ on the debris target.

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In a homogeneous medium, the self-focusing length is expressed as $L_{SF} = k w_{0}^{2} / 2 \sqrt{P / P_{cr} - 1}$ [20], and in the ground homogeneous atmosphere we have L_SF = 197 km when P = 900 P_cr, $w_{0} = \sqrt{2} m$ and $λ = 1 .06 μm$ . Furthermore, the self-focusing length of a laser beam propagating vertically from the ground to the orbit is larger than that in the ground homogeneous atmosphere. Thus, the self-focusing length is much longer than the thickness of the atmosphere when a laser beam with large spot size (e.g., over 1 m) propagates vertically from the ground to the orbit, and then the filamentation doesn’t occur even if P is much high.

The spatial and temporal intensity distributions normalized to the initial peak intensity I_in versus the propagation distance z are given in Figs. 2(a)-2(c) and Figs. 2(d)-2(f), respectively. It is shown that the spatial side lobe and the temporal pulse splitting occur earlier on propagation as P increases. Furthermore, the spatial side lobe and the temporal pulse splitting are usually coupled together. The strong self-focusing moves off-axis energy toward the peak of the pulse and compresses it in both space and time [21]. The self-focusing becomes stronger as the peak intensity increases, which results in appearing of the spatial side lobe. On the other hand, the self-phase modulation (SPM) increases as the peak intensity increases, which results in generating new frequency components [22]. But the GVD effect results in broadening of the pulse duration. The combination of the SPM and GVD results in the pulse splitting. Thus, the temporal pulse splitting appears usually together with the spatial side lobe.

Fig. 2 (a)-(c) spatial distributions and (d)-(f) temporal intensity distributions versus the propagation distance z, T₀ = 2 ps.

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Changes of the on-axis intensity I_axis versus the propagation distance z for different values of T₀ and P are shown in Fig. 3, where I_axis denotes the intensity at r = 0 and t = 0. Figure 3 shows that the position of maximum intensity is farther away from the debris target as P or T₀ increases. The position of maximum intensity is nearer to the debris target for the femtosecond pulse than that for nanosecond and picosecond pulses. Furthermore, for the femtosecond pulse I_axis on the debris target increases as P increases (see Fig. 3(c)), but this situation is inverse for nanosecond and picosecond pulses (see Figs. 3(a) and 3(b)). For different values of P, the difference of the position of maximum intensity becomes smaller for nanosecond, picosecond and femtosecond pulses in sequence.

Fig. 3 Relative pulse width T/T₀ versus the propagation distance z.

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From Fig. 3(c), it can be seen that, for a femtosecond pulse the on-axis intensity I_axis decreases within the initial atmospheric propagation because the GVD effect is very strong, which results in decreasing of self-focusing. Thus, the pulse splitting on the debris target doesn’t appear for a femtosecond pulse just as shown in Fig. 1.

Changes of the relative pulse width I/I₀ versus the propagation distance z for different values of T₀ and P are given in Fig. 4, where the diffraction and the GVD effects are considered for the linear case. On the debris target it is T/T₀> 1, i.e., the pulse width on the debris target increases in comparison with the initial pulse width T₀. On the debris target, the T increases as P increases, and the T increases in comparison with the linear case. It is noted that, in the linear case, the pulse width is almost unchanged with propagation distance z for nanosecond and picosecond pulses (see the black line in Figs. 4(a) and 4(b)), and even for 400 fs pulse that rapidly broadens only at the initial stage of propagation (see the black line in Fig. 4(c)). However, this situation is different (i.e., a noticeable change of the pulse width on propagation) when a powerful laser pulse propagates vertically from the ground to the orbit even if the atmosphere height is only 40 km. The physical reason is the interplay of the self-focusing effect and the GVD effect. The interplay of self-focusing and GVD in the initial 40km atmosphere propagation causes the phase-front aberration. The phase-front aberration is important, and results in deviations from the behavior of the following free-space propagation in the linear case.

Fig. 4 Relative pulse width T/T₀ versus the propagation distance z.

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Changes of the beam width w versus the propagation distance z for different values of T₀ and P are shown in Fig. 5. As compared with the linear case, the w on the debris target increases. As P increases, the w on the debris target increases and the position of the minimum beam width is farther away from the debris target because the self-focusing becomes stronger. However, this situation is inverse as T₀ decreases, i.e., the w on the debris target decreases and the position of the minimum beam width is closer to the debris target as T₀ decreases. The physical reason is that the GVD becomes stronger as T₀ decreases, and the spatiotemporal coupling results in a decrease of the beam width [23].

Fig. 5 Beam width w versus the propagation distance z.

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4. Formulae of the modified focal length and the M²-factor

From Figs. 1 and 2, it can be seen that the beam quality may be distorted in spatial and temporal distributions by self-focusing and GVD effects. In this section, the crucial formulae (e.g., the modified focal length and the M²-factor) for laser debris removal are derived to achieve the good beam quality on the debris target.

It is known that the B integral represents the nonlinear phase shift due to self-focusing. If the constant factor of the B integral is omitted, we have $\int_{0}^{L} n_{2}_{0} \exp (- z / h) d z \approx n_{2}_{0} h$ . It means that the laser beam propagation from the ground to the orbit in the atmosphere can be treated as two stages, i.e., a propagation of the laser beam over a distance h = 6 km in the homogeneous medium with the nonlinear refractive index n₂₀, and followed by a propagation of the laser beam over a L-h distance in free space. Based on Eq. (1) without the group velocity dispersion term, after the first stage propagation in the homogeneous nonlinear medium, the beam width w₁ and the curvature radius R₁ at the plane z = h can be expressed as [24]

w_{1}^{2} = (1 - P / P_{cr}) h^{2} λ^{2} / π^{2} w_{0}^{2} + {(1 - h / F)}^{2} w_{0}^{2},

w_{1}^{2} / R_{1} = (1 - P / P_{cr}) h λ^{2} / π^{2} w_{0}^{2} - (1 - h / F) w_{0}^{2} / F .

For the second stage propagation in free space, the beam width w₂ can be derived as

w_{2}^{2} = w_{1}^{2} + \frac{2 w_{1}^{2}}{R_{1}} (z - h) + (\frac{λ^{2}}{π^{2} w_{1}^{2}} + \frac{w_{1}^{2}}{R_{1}^{2}}) {(z - h)}^{2} .

It is noted that the beam isn’t Gaussian profile exactly due to self-focusing. Thus, Eq. (8) is needed to be revised. According to the numerical simulation results, Eq. (8) is modified as

w_{\mod}^{2} = w_{1}^{2} + \frac{2 w_{1}^{2}}{R_{1}} (z - h) + (\frac{A λ^{2}}{π^{2} w_{1}^{2}} + \frac{w_{1}^{2}}{R_{1}^{2}}) {(z - h)}^{2},

where

A = 1- u (P / P_{cr}) + v {(P / P_{cr})}^{2} > 1

is the modifying factor, and u = 8.5 × 10⁻⁴exp(−1.77w₀), v = 2.01 × 10⁻⁴exp(−3.93w₀). From Fig. 6 one can see that the analytical result by using Eq. (9) is in agreement with the numerical simulation results.

Fig. 6 Comparison of fitting results and numerical simulation results.

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The position of focus moves due to beam self-focusing in the atmosphere. The focus position $F'$ (i.e., the position of the minimum beam width) can be determined by the equation $d w_{_{\mod}}^{2} / d z = 0$ , which is derived as

F' = \frac{1 - h / F}{(1 - h / F) / F + (P / P_{cr} - 1) h λ^{2} / (π^{2} w_{0}^{4})} .

We demonstrate numerically that the focus position is almost the same for a CW laser and a nanosecond laser pulse although the beam width minimums are different for these two cases. From Eq. (10) one can see that F '<F . Thus, the focal length should be modified to guarantee the focus spot on the debris target. Based on Eq. (10), the focal length is modified as

F_{\mod} = \frac{(L + h) + \sqrt{{(L - h)}^{2} + 4 (P / P_{cr} - 1) h^{2} L^{2} λ^{2} / (π^{2} w_{0}^{4})}}{2 [1 - (P / P_{cr} - 1) h L λ^{2} / (π^{2} w_{0}^{4})]} .

Figure 7 shows that F_mod increases as P increases or w₀ decreases. The comparison of intensity distributions on the debris target before and after modification is shown in Fig. 8, where a nanosecond laser pulse is considered. Figure 8 indicates that the beam quality on the debris target becomes better if the modified focal length F_mod = 1.52 × 10³ km is adopted. It means that Eq. (11) is valid approximatively even for nanosecond laser pulses. On the other hand, the optimal pulse duration for the NIF laser system is approximately 4 ns [8]. Therefore, the formula of the modified focal length obtained in this paper is useful for space-debris removal.

Fig. 7 Modified focal length F_modversus the relative power P/P_cr.

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Fig. 8 Comparison of intensity distributions on the debris target, w₀ = 1.41 m, P = 2000 P_cr,T₀ = 4ns.

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The M ²-factor is widely used to characterize the quality of laser radiation. The far-field divergence angle θ and the beam waist w_min can be obtained from Eq. (9), i.e.,

θ = \lim_{z \to \infty} (\frac{w_{\mod}}{z}) = \sqrt{\frac{A λ^{2}}{π^{2} w_{1}^{2}} + \frac{w_{1}^{2}}{R_{1}^{2}}},

w_{\min} = \sqrt{\frac{A λ^{2} / π^{2}}{A λ^{2} / (π^{2} w_{1}^{2}) + w_{1}^{2} / R_{1}^{2}}} .

According to the definition of M ²-factor [18,24] and Eqs. (12) and (13), the analytical expression of M ²-factor of laser beams for ground-based debris removal can be obtained, i.e.,

M^{2} = \frac{π}{λ} θ w_{\min} = \sqrt{A} .

The intensity distributions on the debris target for different values of M ², w₀ and P are shown in Fig. 9, where nanosecond laser pulses are considered. Figure 9 indicates that the beam profile is similar when the value of M ²-factor is the same whether the values of w₀ and P. It implies that Eq. (14) is valid approximatively for nanosecond laser pulses. In particular, it is found that the beam quality on the debris target is good if the value of M ²-factor is less than 1.6 (see Figs. 9(a) and 9(b)). It is noted that this result is not valid if the initial field is not an ideal Gaussian field.

Fig. 9 Intensity distribution on the debris target for different values of M², w₀ and P, T₀ = 4 ns.

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Figure 10 shows the curve of M ² = 1.6 for different w₀ and P/P_cr. It is clear that the values of w₀ and P/P_cr must be adopted below this curve (i.e., the green area) to achieve the good beam quality on the debris target. Figure 10 also indicates that the value of M ²-factor decreases as w₀ increases or P/P_cr decreases.

Fig. 10 Curve of M ² = 1.6 for different values of w₀ and P/P_cr.

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5. Conclusions

In this paper, we study self-focusing and GVD effects in the inhomogeneous atmosphere on pulsed-laser space debris removal facilitated by a ground-based laser. It is found that changes of the pulse duration and the beam spot size with the propagation distance are noticeable due to the interplay of the GVD effect and the self-focusing effect, which is quite different from the behavior in the linear case. It is shown that the temporal pulse splitting may appear on the debris target for nanosecond and picosecond pulses, and the spatial side lobe usually appears together with the temporal pulse splitting. The position of maximum intensity is nearer to the debris target for the femtosecond pulse than that for nanosecond and picosecond pulses. As compared with the linear case, the beam width and pulse width on the debris target increase. On the other hand, crucial formulae of the modified focal length and the M ²-factor for ground-based debris removal are also derived in this paper. It is found that the beam quality on the debris target becomes better if our modified focal length is adopted, and the beam quality on the debris target will be good if the value of M ²-factor is less than 1.6. The results obtained in this paper are very useful for space-debris removal facilitated by a ground-based pulsed laser.

Funding

National Natural Science Foundation of China (NSFC) (61775152, 61475105, 61505130).

References and links

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Self-focusing and group-velocity dispersion of pulsed laser beams in the inhomogeneous atmosphere

Abstract

1. Introduction

2. Theoretical model

3. Temporal and spatial intensity distributions

4. Formulae of the modified focal length and the M²-factor

5. Conclusions

Funding

References and links

Cited By

Figures (10)

Equations (14)

Optics Express

Abstract

1. Introduction

2. Theoretical model

3. Temporal and spatial intensity distributions

4. Formulae of the modified focal length and the M2-factor

5. Conclusions

Funding

References and links

Cited By

Figures (10)

Equations (14)

Optics Express

4. Formulae of the modified focal length and the M²-factor