Uniform irradiation generated by beam self-focusing in the inhomogeneous atmosphere

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

doi:10.1364/OE.27.014585

1. Introduction

A great number of space debris seriously threaten the safety of spacecrafts [1]. The space debris removal using a high-power ground-based laser is one of the main methods for this problem [2,3]. The space-debris removal facilitated by a ground-based pulsed laser was analyzed by ‘Orion’ project [4]. The laser power for space-debris clearing is well above the critical power for self-focusing in air [5]. It has been proved that the beam filamentation caused by self-focusing can be avoided because the earth’s atmosphere is inhomogeneous [5–7]. Recently, Rubenchik et al. demonstrated that the self-focusing can noticeably decrease the laser intensity on the debris target [5], and Vaseva et al. proposed that it may be possible to eliminate the impact of self-focusing in the atmosphere on the laser beam by using adaptive optics [8]. Very recently, our group showed that the temporal pulse splitting may appear on the debris target, and the spatial side lobe usually appears together with the temporal pulse splitting [9].

The uniform irradiation on the target is important for some applications (e.g., inertial confinement fusion and laser materials processing) [10–13]. The uniform irradiation on the target is also interesting for the laser space-debris clearing. It is known that the flat-topped profile will change when a flat-topped beam propagates in free space (i.e., linear propagation). Furthermore, we have shown that the flat-topped profile will also change for nonlinear propagation in the homogeneous atmosphere because the beam self-focusing effect results in two intensity peaks. However, in this paper we demonstrate that this situation is different when a flat-topped beam propagates from the ground through the inhomogeneous atmosphere to space orbits. The uniform irradiation on the debris target may achieve for an initial flat-topped beam because of the phase modulation caused by self-focusing in the inhomogeneous atmosphere and the diffraction beyond the atmosphere. On the other hand, Rubenchik et al. proposed that initial beam defocusing can compensate for the detrimental effect of nonlinearity [5]. In this paper, we show that the Kerr nonlinearity in the inhomogeneous atmosphere can be decreased greatly if an initial flat-top beam was focused with focus length chosen based on the orbit height, which results in increasing laser intensity and achieving good beam quality on the debris target.

2. Theoretical model

We consider that a flat-topped laser beam propagates along the vertical direction z from the ground to debris targets. Under the standard paraxial approximation, the beam features of diffraction and Kerr nonlinearity can be described by the nonlinear Schrödinger equation (NLS) [5], i.e.,

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

where A is the envelope of the electric field, k₀ is the wave-number related to the wave length λ by k₀ = 2π/λ,

\nabla_{⊥}^{2} = \partial^{2} / \partial r^{2} + (1 / r) \partial / \partial r,

n₀ and n₂ are linear and nonlinear refractive indexes respectively. In the atmosphere, n₂ is a function of altitude, i.e.,

n_{2} (z) = n_{20} \exp [- (z / h)],

where h = 6 km, and n₂₀ = 4.2 × 10⁻¹⁹ cm²/W is the refractive index on the ground [5]. In this paper, the orbit height is adopted as 1000km where the debris is most concentrated [14], and the self-focusing effect is considered only within 20km from the sea level because n₂ decreases with increasing z. In addition, the intensity I is related to the envelope of the electric field A by I = A*A.

It is difficult to obtain the analytical result of Eq. (1), but numerical results can be generated. The multi-phase screen method can be applied to solve Eq. (1) [5]. Let A(r, z_j) be the solution of Eq. (1) at z_j plane, and the solution of Eq. (1) at z_j₊₁ = z_j + Δz plane can be written as [15]

A (r, z_{j + 1}) = \exp (- \frac{i}{4 k_{0}} Δ z \nabla_{⊥}^{2}) \exp (- i s) \exp (- \frac{i}{4 k_{0}} Δ z \nabla_{⊥}^{2}) A (r, z_{j}),

where s is the phase modulation due to the Kerr nonlinearity effect within Δz nonlinear propagation. Equation (2) indicates that the propagating field over a distance Δz in a nonlinear medium consists of three steps, i.e., a free space propagation of the field over a distance Δz/2, then an incrementing of the phase caused by the Kerr nonlinearity effect within Δz nonlinear propagation, and last a free space propagation of the resulting field over a distance Δz/2. Thus, the beam propagation through a nonlinear medium can be studied by using the multi-phase screen method.

When the beam diffraction is ignored, Eq. (1) reduces to

i \frac{\partial A}{\partial z} + k_{0} \frac{n_{2}}{n_{0}} | A^{2} | A = 0.

When a laser beam propagates in a nonlinear medium from z_j to z_j + Δz, the solution of Eq. (3) can be derived as

A (r, z_{j} + Δ z) = A (r, z_{j}) \exp [\frac{i k_{0} n_{2} {| A (r, z_{j}) |}^{2}}{n_{0}} Δ z] .

It is noted that Eq. (4) is obtained approximatively if

{| A (r, z) |}^{2}

is a constant, because the change of

{| A (r, z) |}^{2}

is very small during propagation over a distance Δz. From Eq. (4), we can obtain the phase modulation due to the Kerr nonlinearity effect within Δz nonlinear propagation, i.e.,

s = k_{0} n_{2} {| A (r, z_{j}) |}^{2} Δ z / n_{0} .

The electric field of flat-topped beams at the plane z = 0 can be expressed as [16]:

A (r, z = 0) = A_{0} \exp (- \frac{i C_{0}}{w_{0}^{2}} r^{2}) \sum_{m = 1}^{N} \frac{{(- 1)}^{m - 1}}{N} (\begin{matrix} N \\ m \end{matrix}) \exp (- \frac{m r^{2}}{w_{0}^{2}}),

where N is the beam order, m is a positive integer. Equation (5) reduces to the expression of Gaussian beams when N = 1. P is the beam power, w₀ is the initial beam radius,

(\begin{matrix} N \\ m \end{matrix})

is the binomial coefficient, C₀ = kw₀²/2F, and F is the focal length. The expression of A₀ can be obtained by using the relationship between the beam power P and the electric field A (i.e.,

P = \int_{0}^{2 π} d θ \int_{0}^{\infty} {| A |}^{2} r d r = const

), which can be written as

A_{0} = \sqrt{P / π w_{0}^{2} \sum_{m}^{N} \sum_{n}^{N} \frac{{(- 1)}^{m + n}}{N^{2}} (\begin{matrix} N \\ m \end{matrix}) (\begin{matrix} N \\ n \end{matrix}) \frac{1}{m + n}} .

By using the multi-phase screen method and the discrete Fourier transform method [15,17], we design a computer code of propagation of a flat-topped laser beam along the vertical direction z from the ground to the debris target. In this paper, the numerical calculation parameters are taken as λ = 1.06μm, F = 1000km, C₀ = 5.93 and $w_{0} = \sqrt{2} m$ . In addition, the Gaussian beam critical power P_cr = λ²/(2πn₀n₂₀) = 4.3GW is adopted as the normalized factor to the beam power. The initial 3D intensity distributions of flat-topped beams with different order N are shown in Fig. 1, where r = (x, y). As N increases, the value of the peak intensity decreases, but a flatter-topped beam appears. It is noted that the beam power should be corresponding to the 3D intensity volume. In fact, it is the same beam power for different beam order N in Fig. 1.

Fig. 1 Initial 3D intensity distributions of flat-topped beams with different beam order N, P = 1400P_cr.

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3. Influence of the beam order on the beam quality due to self-focusing

In this paper, the mean square beam width is adopted to describe the beam spot size on debris targets, which is defined as [18]:

w^{2} = \frac{2}{P} \int_{0}^{2 π} \int_{0}^{\infty} r^{2} A (r, z) A^{*} (r, z) r d r d θ .

The changes of beam width w_lin for linear propagation in free space and beam width w for nonlinear propagation in the atmosphere versus the propagation distance z are shown in Figs. 2(a) and 2(b), respectively. It can be seen that the beam spot size on the debris target decreases as N increases. In comparison with the linear propagation (see Fig. 2(a)), in the atmosphere the position of the beam width minimum is far away from the debris target due to self-focusing effect, and is further away from the debris target as N decreases (see Fig. 2(b)). Thus, the beam spot size on the debris target increases due to self-focusing effect. The relative beam width w/w_lin versus the propagation distance z is shown in Fig. 3. It can be seen that, w/w_lin near the debris target increases rapidly as N decreases, i.e., the influence of self-focusing on the beam spot size decreases rapidly as N increases. Therefore, the much smaller beam spot size on the debris target may achieve for an initial flat-topped beam than that for an initial Gaussian one.

Fig. 2 (a) Beam width w_lin for linear propagation in free space and (b) beam width w for nonlinear propagation in the atmosphere versus the propagation distance z, P = 1400P_cr.

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Fig. 3 Relative beam width w/ w_lin versus the propagation distance z, P = 1400P_cr.

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The changes of the on-axis intensity I(r = 0, z) versus the propagation distance z are shown in Fig. 4. It can be seen that, in the atmosphere the position of the maximum of I(r = 0, z) is far away from the debris target due to self-focusing effect as compared with that for the linear propagation in free space. Furthermore, the influence of self-focusing on the I(r = 0, z) decreases as N increases. In particular, the I(r = 0, z) on the debris target is much larger for an initial flat-topped beam than that for an initial Gaussian one.

Fig. 4 On-axis intensity I(r = 0, z) versus the propagation distance z, P = 1400P_cr.

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The relative intensity distributions I(r, z)/I₀ versus the propagation distance z are shown in Fig. 5, where I₀ is the peak intensity on the debris target for the linear propagation in free space. If a high beam power is adopted (e.g., P = 2250P_cr in Fig. 5), the beam distortion on the debris target appears for an initial Gaussian beam (see Fig. 5(a)), but the good beam quality on the debris target achieves for an initial flat-topped beam (see Fig. 5(c)).

Fig. 5 Relative intensity distribution I(r, z)/I₀ versus the propagation distance z, P = 2250P_cr.

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The B integral is an important characteristic parameter to quantitatively describe the beam quality degradation due to self-focusing effect. The value of the B integral must be below 3-4 in order to avoid filamentation [5]. The B integral is defined as [19]

B = k_{0} \int_{0}^{z_{0}} I (r = 0, z) n_{2} (z) d z .

where z₀ is the thickness of the atmosphere (z₀ = 20km is adopted in this paper). It is noted that, for the situation considered in this paper (i.e., F = 1000km), the I(r = 0, z) almost doesn’t change within the 20km propagation in the atmosphere. Therefore, in Eq. (8), the I(r = 0, z) can be replaced by the I(r = 0, z = 0), and the expression of I(r = 0, z = 0) can be obtained from Eq. (6). Thus, the expression of the B integral of a flat-topped laser beam propagating in the atmosphere from the ground to the debris target can be derived as:

B = \frac{k_{0} n_{20} P h [1 - \exp (- z_{0} / h)]}{π w_{0}^{2}} \frac{\sum_{m = 1}^{N} \sum_{n = 1}^{N} {(- 1)}^{m + n} (\begin{matrix} N \\ m \end{matrix}) (\begin{matrix} N \\ n \end{matrix})}{\sum_{m = 1}^{N} \sum_{n = 1}^{N} {(- 1)}^{m + n} (\begin{matrix} N \\ m \end{matrix}) (\begin{matrix} N \\ n \end{matrix}) \frac{1}{m + n}} .

The changes of the B integral versus the beam order N are shown in Fig. 6. It can be seen that for a particular value of beam power P, the B integral decreases as N increases. Therefore, the self-focusing effect becomes weaker as N increases, and the self-focusing effect for an initial flat-topped beam is weaker than that for an initial Gaussian one, which is in agreement with the results obtained above.

Fig. 6 B integral versus the beam order N, P = 1500P_cr.

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4. Uniform irradiation on the debris target

In the section 3, it has been proved that, the good beam quality on the debris target may achieve for an initial flat-topped beam even if the beam power is very high, but it is unavailable for an initial Gaussian beam (i.e., the beam distortion occurs). In this section, it is shown that the uniform irradiation (i.e., the flat-topped beam profile) on the debris target may also achieve for an initial flat-topped beam if the beam power is high enough (see Fig. 7). In particular, we have demonstrated that the uniform irradiation does not achieve for linear propagation in free space or for nonlinear propagation in the homogeneous atmosphere.

Fig. 7 Intensity distribution I(r, z) versus the propagation distance z. Uniform irradiation on the debris target.

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The intensity distribution I(r, z) and the phase distribution Ф(r, z) at the end of the inhomogeneous atmosphere (z = 20km) are shown in Figs. 8(a) and 8(b), respectively. The relative intensity distribution I(r, z)/I_pea on the debris target is shown in Fig. 8(c), where I_pea is the peak intensity on the debris target. It is noted that the phase modulation caused by the self-focusing in the inhomogeneous atmosphere nearly doesn’t affect the intensity distribution in the near field (see Fig. 8(a)), but it will obviously affect the intensity distribution in the far field (see Fig. 8(c)). In addition, the power of a flat-topped beam is concentrated around the center axis. Thus, the phase modulation near the center axis plays an important role. Near the center axis, the phase distribution for nonlinear propagation in the inhomogeneous atmosphere is quite different from that for linear propagation in free space (see Fig. 8(b)). This is the main physical reason why the uniform irradiation (i.e., the flat-topped beam profile) on the debris target may appear (see Fig. 8(c)).

Fig. 8 (a) intensity distribution I(r, z) and (b) phase distribution Ф(r, z) at the plane z = 20km, (c) relative intensity distribution I(r, z)/I_pea on the debris target (z = 1000km), N = 10, P = 8050P_cr.

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To achieve the uniform irradiation on the debris target, the beam power P must be high and change with the beam order N. The required beam power P_fla for the uniform irradiation on the debris target versus the beam order N is shown in Fig. 9(a). From Fig. 9(a), it can be seen that P_fla increases as N increases. Figure 9(b) shows the corresponding B integral (i.e., B_fla) for the uniform irradiation on the debris target. One can see that, to achieve the uniform irradiation on the debris target, the B_fla is approximately equal to 3.5 for different values of N, where the value of P_fla changes as N changes (see Fig. 9(a)). It means that, only when the phase modulation caused by the self-focusing effect reaches a certain degree, the uniform irradiation on the debris target may appear.

Fig. 9 For the uniform irradiation on the debris target, (a) required beam power P_fla and (b) B integral B_fla versus the beam order N, F = 1000km.

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When the focal length is fixed, the uniform irradiation appearing is dependent on both the phase modulation caused by self-focusing in the inhomogeneous atmosphere and the diffraction beyond the atmosphere, and the uniform irradiation usually doesn't appear at the debris target. However, the uniform irradiation on the debris target may achieve by modifying the focal length. The changes of the modified focal length F_mod for the uniform irradiation on the debris target versus the beam power P and the beam order N are shown in Fig. 10(a) (see the black dots in Fig. 10(a)). One can see that F_mod increases as P increases and N decreases. Based on the black dots in Fig. 10(a), we obtain the fitting formula of the F_mod for the uniform irradiation on the debris target, i.e.,

F_{\mod} = \frac{F_{0} + A_{01} N {+B}_{01} (P / P_{cr}) + B_{02} {(P / P_{cr})}^{2} + B_{03} {(P / P_{cr})}^{3}}{1 + A_{1} N + A_{2} N^{2} + A_{3} N^{3} + B_{1} (P / P_{cr}) + B_{2} {(P / P_{cr})}^{2}},

where the coefficients are shown in Table 1. Based on Eq. (10), the fitting surface is also given in Fig. 10(a). one can see that the fitting surface is in agreement with the black dots. The fitting model can be examined with the Adjusted R-Squared (R²_Adj), which is a measure of the amount of variation around the mean explained by the model [20]. It is shown that R²_Adj = 0.997 for our fitting Eq. (10), which is close to value 1. It means our fitting Eq. (10) is reliable and simple, which presents an effective design rule for the uniform irradiation on the debris target.

Fig. 10 For the uniform irradiation on the debris target, (a) modified focal length F_mod versus the beam order N and the beam power P (black dots: numerical simulation results; Curve surface: fitting surface by using Eq. (10)), and (b) the range of N and P in Eq. (10).

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Table 1. Values of the coefficients in Eq. (10)

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It is mentioned that the uniform irradiation on the debris target cannot appear when the beam order N is less than 5 or the beam power is not high enough because the phase modulation caused by the self-focusing cannot reach a certain degree. For example, when N = 10 and F = 1000km, the uniform irradiation cannot appear on the debris target if the beam power is smaller than 8050Pcr because of B < 3.5. Figure 10(b) shows that the range of N and P for the uniform irradiation on the debris target, i.e., the green area is available.

5. Conclusions

In this paper, the uniform irradiation on the target for the ground-based space-debris clearing is studied. For an initial flat-topped beam, the flat-topped profile will change for linear propagation in free space or for nonlinear propagation in the homogeneous atmosphere. However, it is found that the flat-topped profile may appear on propagation of an initial flat-topped beam because of the phase modulation caused by self-focusing in the inhomogeneous atmosphere and the diffraction beyond the atmosphere. This may offer a way to achieve the uniform irradiation under the effect of inhomogeneous nonlinearity, which is also useful for other applications (e.g., inertial confinement fusion and laser materials processing).

To achieve the uniform irradiation on the debris target, the fitting formula of the modified focal length is also presented in this paper, which presents an effective design rule for the uniform irradiation on the debris target. On the other hand, it is shown that the good beam quality on the debris target may achieve for an initial flat-topped beam even if the beam power is very high, but it is unavailable for an initial Gaussian beam (i.e., the beam distortion occurs). Under the good beam quality condition, the much smaller beam spot size and much larger intensity on the debris target may achieve for an initial flat-topped beam than that for an initial Gaussian one. The results obtained in this paper are theoretical and practical interest.

Funding

National Natural Science Foundation of China (NSFC) (61775152).

References

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Coefficient	F₀ (km)	A₀₁ (km)	B₀₁(km)	B₀₂(km)	B₀₃(km)
Value	818.69447	−22.59163	0.11577	−1.01252E-5	1.10179E-10
Coefficient	A₁	A₂	A₃	B₁	B₂
Value	0.00849	−0.00167	3.72142E-5	3.95178E-5	−5.38818E-9

Uniform irradiation generated by beam self-focusing in the inhomogeneous atmosphere

Abstract

1. Introduction

2. Theoretical model

3. Influence of the beam order on the beam quality due to self-focusing

4. Uniform irradiation on the debris target

5. Conclusions

Funding

References

Cited By

Figures (10)

Tables (1)

Equations (10)

Optics Express