Changes of the centroid position of laser beams propagating through an optical system in turbulent atmosphere

Optics & Laser Technology 54 (2013) 199–207

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Changes of the centroid position of laser beams propagating through an optical system in turbulent atmosphere Xiaoling Ji a,n, Yahya Baykal b, Xinhong Jia a a b

Department of Physics, Sichuan Normal University, Chengdu 610066, People's Republic of China Department of Electronic and Communication Engineering, Çankaya University, Eskişehir yolu, 29. km. 06810 Yenimahalle, Ankara, Turkey

art ic l e i nf o

a b s t r a c t

Article history: Received 14 December 2012 Received in revised form 7 May 2013 Accepted 24 May 2013 Available online 21 June 2013

In this paper, the effects of atmospheric turbulence, initial field amplitude, optical system and thermal blooming on the centroid position of laser beams propagating through the atmosphere are studied in detail. With the average over the ensemble of the turbulent medium, the centroid position is independent of turbulence. However, the centroid position depends on the centroid positions at the source plane and in the far-field, and the elements of ray-transfer-matrix. The physical reason why the centroid position changes on propagation is that the far-field centroid position is not located on the propagation z-axis due to the field phase distortion and the decentred intensity. The centroid position of laser beams with the spherical aberration and the decentred intensity is examined analytically. When laser beams with the decentred intensity propagate through the atmosphere, the effect of thermal blooming on the centroid position is investigated by using the four-dimensional (4D) computer code of the time-dependent propagation of high power laser beams through the atmosphere. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Centroid position Turbulent atmosphere Laser beams with distortion

1. Introduction The propagation of laser beams in different media is a topic that has been of considerable theoretical and practical interest for a long time [1–12]. The centroid position (or the centre of gravity [4]) of centred and aligned laser beams is located on the z-axis during propagation. However, the propagation of the decentred and the misaligned laser beams is often encountered. For example, highpower laser beams produced by unstable optical resonators always possess decentred field (e.g., Fig. 1). The misaligned beams appear when laser beams propagate through misaligned optical systems. The effect of the misaligned optical systems on the centroid position can be treated as laser beams having tilt aberration. Theoretically, higher-order moments are very useful for the characterisation of partially coherent beams. The first-order moment indicates the centroid position. For the decentred and the misaligned laser beams, higher-order moments about the centroid position are more interesting than those about the origin of coordinates. In recent years, higher-order moments of partially coherent beams, super-Gauss beams, and general truncated beams propagating in turbulence were investigated in [13–15], respectively. The propagation of higher-order moments in these studies [13–15] are about the origin of coordinates because their first-order moments are zero. The change of centroid position on propagation is very important for some practical applications, such as remote sensing, imaging and communication systems, high-energy weapons. In 1992, the propagation of the first-order n

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moments in quadratic-index media was studied in [4]. Airy beams with decentred field have several properties that make them of interest in modern optics. Recently, higher-order moments of airy beams about the centroid position were studied in [16]. Very recently, we studied the propagation of partially coherent annular beams with decentred field in turbulence along a slant path [17], where the effects of the optical systems, aberrations and thermal blooming on the centroid position were not considered. The aim of this paper is to study the changes of the centroid position of laser beams propagating through an optical system in turbulent atmosphere. In Sections 2 and 3, the effects of the initial field amplitude, optical system and atmospheric turbulence on the centroid position are examined in detail, where the laser beams with spherical and tilt aberrations are considered. In Section 4, the effect of thermal blooming on the centroid position of laser beams propagating through the atmosphere is investigated by using the four-dimensional (4D) computer code of the time-dependent propagation of high power laser beams through the atmosphere, where the laser beam power, the time, the atmosphere absorption coefficient and the cross wind are considered. The physical explanations for main results obtained in this paper are given in Section 5. In Section 6, the results obtained in this paper are summarised.

2. Formula of the centroid position In this and the next sections, the effect of thermal blooming is not considered. Based on the extended Huygens–Fresnel principle,

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the cross-spectral density function of a partially coherent beam propagating through an ABCD optical system in atmospheric turbulence can be expressed as [2,18]
2 Z ∞ Z ∞ k Wðρ1 ; ρ2 ; LÞ ¼ dρ′1 dρ′2 Wðρ′1 ; ρ′2 ; 0Þ 2πB −∞ −∞ 
h i ik Aðρ′1 2 −ρ′2 2 Þ−2ðρ1 ρ′1 −ρ2 ρ′2 Þ þ Dðρ21 −ρ22 Þ exp 2B  ð1Þ −Hðρ1 ; ρ2 ; ρ′1 ; ρ′2 ; LÞ where Wðρ′1 ; ρ′2 ; 0Þ is the cross-spectral density function at the source plane z¼ 0, L denotes the distance between the source and the receiver planes, ρ′≡ðx′; y′Þ and ρ≡ðx; yÞ are the transverse vectors at the source plane z¼ 0 and the receiver plane z¼ L, respectively. A, B and D are the elements of the ray-transfer-matrix for the optical system between the source and the receiver planes, and k is the wave number related to the wave length λ by k ¼2π⧸λ. The last term in Eq.(1), i.e., exp[−H] represents the effect of the turbulence. With the average over the ensemble of the turbulent medium, H can be expressed as [18] Z LZ ∞   2 H ρ1 ; ρ2 ; ρ′1 ; ρ′2 ; L ¼ 4π 2 k κΦn ðκ; zÞ 0 0

  bðzÞ  bðzÞ   ðρ′1 −ρ′2 Þ þ aðzÞ− ðρ1 −ρ2 Þ dκdz ð2Þ 1−J 0 ðκ  B B where a(z) and b(z) are the ray-transfer-matrix elements for backward propagation through the optical system (i.e., from the receiver plane to a propagation distance z). J 0 ðÞ is the zero-order Bessel function of the first kind, κ is the magnitude of spatial wave number, and Φn ðκ; zÞ is the spatial power spectrum of the refractive-index fluctuations of the turbulent medium. Based on the definition of the first-order moment of the Wigner distribution function (WDF) [19,20], and following the treatment in Ref. [13,14], we can obtain the expressions for the centroid position and the far-field divergence angle of the partially coherent beam propagating through an ABCD optical system in atmospheric turbulence, which are given by    hxi ¼ Ahx′i þ Bhθ′x i; y ¼ A y′ þ B θ′y ð3Þ    hθx i ¼ C hx′i þ Dhθ′x i; θy ¼ C y′ þ D θ′y ð4Þ  where hx′i and y′ represent the centroid position along  the x-axis and y-axis at the source plane, respectively; hθ′x i and θ′y denote the far-field divergence angle of the initial field along the x-axis and y-axis, respectively. Physically, hθ′x i and θ′y can also be regarded as

the far-field centroid positions of the initial field along the x-axis and y-axis, respectively. Eq. (3) indicates that the centroid positions are independent of turbulence, but depend on the centroid positions at the source plane and in the far-field, and the elements of the raytransfer-matrix. It is mentioned that Eqs. (3) and (4) are valid under the average over the ensemble of the turbulent medium, i.e., they are not instantaneous centroid positions. The instantaneous centroid positions will change versus the time and the strength  of turbulence. In this paper, only centroid positions hxi and y are examined. h i From Eq. (3) we can see that the law of propagation of x is the same as that of y . Therefore, in the following part of Sections 2 and 3 we only study the propagation of hxi. Three interesting optical systems are further discussed as follows. Free space For free space propagation, we have A ¼1, B ¼L. And then, Eq. (3) reduces to hxi ¼ hx′i þ Lhθ′x i

ð5Þ

Thin lens At the focal plane of a thin lens, we have A¼0. Then, Eq. (3) reduces to hxi ¼ Bhθ′x i

ð6Þ

Eq. (6) indicates that hxi at the focal plane is independent of hx′i. Imaging optical system At the imaging plane of an optical system, we have B ¼0. Then, Eq. (3) reduces to hxi ¼ Ahx′i ð7Þ Eq. (7) shows that hxi is independent of hθ′x i, and is A times hx′i. It is known that hx′i only depends on the intensity distribution at the source plane, and hx′i becomes zero due to the initial centred intensity. However, the far-field centroid position hθ′x i depends on the field amplitude at the source plane, i.e., both initial intensity and phase, which is discussed in detail in the following Section 3.

3. Discussions about the far-field centroid position At the source plane, the initial field amplitude can be written as Eðx′; 0Þ ¼ E0 ðx′Þexp½−ikϕðx′Þ

ð8Þ

where E0 ðx′Þ and ϕðx′Þ are real functions, E0 ðx′Þ dominates the intensity distribution at the source plane, ϕðx′Þ is the initial field phase.

Fig. 1. (a) 3D intensity distribution Iðx′; y′; 0Þ, and (b) contour lines at the source plane.

Fig. 2. (a) 3D intensity distribution Iðx; y; zÞ, and (b) contour lines at the focal plane in vacuum.

Fig. 3. (a) Contour lines of Iðx′; y′; 0Þ at the source plane, and (b)–(e) contour lines of I(x, y, z) at the focal plane for different values of time t when P ¼60 kW, v¼ 0 and α ¼ 1:252  10−5 =m. (a) , (b) t=0s, (c) t=0.2s, (d) t=0.4s, (e) t=0.6s, (f) t=0.8s, (g) t=1s.

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According to Ref. [4], the far-field centroid position can be expressed as hθ′x i ¼ −

i 2Pk

Z

∞

Eðx′; 0Þ −∞

∂En ðx′; 0Þ dx′ þ c:c: ∂x′

ð9Þ

R∞ where P ¼ −∞ Iðx′; 0Þdx′ is the total power of the laser beam, and I ðx′; 0Þ ¼ E20 ðx′Þ is the intensity at the source plane; “c.c.” denotes the conjugate function of the first term in Eq. (9). Substituting Eq. (8) into Eq. (9), we obtain hθ′x i ¼ −

i 2Pk

Z

∞ −∞

ikI ðx′; 0Þ

dϕðx′Þ 1 dIðx′; 0Þ þ dx′ þ c:c dx′ 2 dx′

ð10Þ

Fig. 4. Change of the centroid position at the focal plane versus time t when P ¼60 kW and v¼ 0.

It is assumed that the field is limited to finite space, namely, we have I ðx′; 0Þ ¼ 0 when x′-∞ or x′-−∞. And then, the second term R∞ in Eq. (10) is zero, i.e., −∞ ½dIðx′; 0Þ= dx′ dx′ ¼ 0. Therefore, the far-field centroid position can be rewritten as

Z 1 ∞ dϕðx′Þ hθ′x i ¼ dx′ ð11Þ I ðx′; 0Þ P −∞ dx′ Eq. (11) indicates that hθ′x i is independent of the wave number k. For laser beams without the field phase distortion (i.e., dϕðx′Þ= dx′ ¼ 0), Eq. (11) reduces to hθ′x i ¼ 0

ð12Þ

Fig. 6. Change of the centroid position at the focal plane versus laser beam power P when t ¼1 s, v¼ 0 and α ¼ 1:252  10−5 =m.

Fig. 5. Contour lines of Iðx; y; zÞ at the focal plane for different values of laser beam power P when t ¼1 s, v¼ 0 and α ¼ 1:252  10−5 =m. (a) p=60kW (b) p=30kW (c) p=10kW (d) p=2kW (e) p=0.2kW.

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Fig. 7. Contour lines of Iðx; y; zÞ at the focal plane for different values of the absorption coefficient α when at t¼ 0.2 s, v¼ 0 and P¼60 kW. (a) α=6.5  10-5/m (b) α=4.5  10-5/m (c) α=2.5  10-5/m (d) α=1.5  10-5/m (e) α=0.5  10-5/m (f) α=0.01  10-5/m.

function (i.e., ϕðx′Þ is an even function), but this situation reverses if ϕðx′Þ is an odd function. The laser beams with spherical and tilt aberrations are considered below. Spherical aberration The field phase of laser beams with spherical aberration can be expressed as ϕðx′Þ ¼ C 4 x′4 where C4 is the spherical aberration coefficient. Substituting Eq. (13) into Eq. (11), we have hθ′x i ¼

Fig. 8. Change of the centroid position at the focal plane versus the absorption coefficient α when t ¼0.2 s, v¼ 0 and P¼ 60 kW.

Z

4C 4 P

∞

−∞

x′3 I ðx′; 0Þ dx′ ¼ 0

3.1. Centred intensity distribution For the centred intensity distribution case, I ðx′Þ is an even function. From Eq. (11) we have hθ′x i ¼ 0 if dϕðx′Þ= dx′ is an odd

ð14Þ

Tilt aberration The field phase of laser beams having tilt aberration can be written as ϕðx′Þ ¼ C 1 x′

Eq. (12) together with Eq. (5) indicate that in free space for a laser beam without field phase distortion, the centroid position does not change on propagation, which is the same as that at the source plane. From Eq. (12) together with Eq. (6) we can conclude that for a laser beam without field phase distortion, at the focal plane the centroid position is located on the propagation z-axis though it has decentred intensity at the source plane. The far-field centroid position of the laser beams having field phase distortion is further discussed as follows.

ð13Þ

ð15Þ

where C1 is the tilt aberration coefficient. Substituting Eq. (15) into Eq. (11), we obtain hθ′x i ¼

C1 P

Z

∞

−∞

I ðx′; 0Þ dx′ ¼ C 1

ð16Þ

3.2. Decentred intensity distribution For the decentred intensity distribution case, I ðx′Þ is not an even or odd function. The E0 ðx′Þ is given by the expression [21,22], ! M x′2 E0 ðx′Þ ¼ ð1−βx′Þ ∑ αm exp −mpm 2 ð17Þ w0 m¼1

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where β is called the decentred parameter,w0 is the waist width, M is beam order, and αm ¼ ð−1Þ

mþ1

M α M! m ;p ¼ ∑ m!ðM−mÞ! m m ¼ 1 m

P¼

M pffiffiffi π w0 ∑

ð18Þ

Spherical aberration Substituting Eqs. (13) and (17) into Eq. (11), after tedious but straightforward integral calculations we obtain pffiffiffi M 6 π w50 βC 4 M αm1 αm2 hθ′x i ¼ − ∑ ∑  5=2 ; P m1 ¼ 1 m2 ¼ 1 m1 pm þ m2 pm 1 2

where

m1 ¼ 1 m2 ¼ 1

"  #  2 m1 pm1 þ m2 pm2 þ w20 β2  3=2 2 m1 pm1 þ m2 pm2

ð20Þ

Eq. (19) together with Eq. (20) indicate that hθ′x i increases linearly as the spherical aberration coefficient C4 increases, and depends on the decentred parameter β. Tilt aberration Substituting Eqs. (15) and (17) into Eq. (11), we obtain hθ′x i ¼ C 1

ð19Þ

M

∑ αm1 αm2

ð21Þ

Eq. (21) shows that hθ′x i is equal to C1, i.e., is independent of β.

Fig. 9. (a) Contour lines of Iðx′; y′; 0Þ at the source plane, and (b)–(e) contour lines of I(x, y, z) at the focal plane for different values of cross wind velocity vx along positive x-axis when t ¼0.4 s, P¼ 60 kW and α ¼ 1:252  10−5 =m. (a), (b) vx=0m/s, (c) vx=0.5m/s, (d) vx=1m/s, (e) vx=3m/s, (f) vx=6m/s, (g) vx=10m/s.

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The initial field of the laser beam with decentred intensity is assumed to be   Γ N þ 1; ðN þ 1Þðx′2 þ y′2 Þ=2a2 Eðx′; y′; 0Þ ¼ ð1−βx′Þ ð25Þ Γ ðN þ 1Þ

Fig. 10. Change of centroid position at the focal plane versus cross wind velocity vx along positive x-axis when t¼ 0.4 s, P ¼60 kW and α ¼ 1:252  10−5 =m.

Compared with Eq. (21) and (16) we can conclude that hθ′x i ¼ C 1 is valid for laser beams with tilt aberration whether the intensity distribution is centred or decentred, which is quite different from the behaviour of that for the spherical aberration case [see Eqs. (14) and (19)]. 4. Effect of thermal blooming on the centroid position In this section the effect of thermal blooming on the centroid position of laser beams propagating through the atmosphere is examined. With the condition of the slowly-varying envelope (SVE) approximation, Maxwell's wave equation is expressed as [3] 2ik

∂E 2 ¼ ∇2⊥ E þ k δεE ∂z

ð22Þ

where ∇⊥2 ¼ ∂2 =∂x2 þ ∂2 =∂y2 , δε ¼ n2 =n20 −1 is the hydrodynamically induced change in permittivity, n is the refractive index, n0 is the refractive index without perturbation, E is a slowly varying field amplitude. It is noted that the retarded time is dropped in Eq. (22) for convenience. The intensity I is given in terms of E, i.e., I ¼ jEj2 expð−αzÞ, where α is the absorption coefficient. Letting En ðx; yÞ be the complete solution to Eq. (22) at z ¼ zn , the solution at z ¼ zn þ Δz may be written as [3] !

 Z n i ik z þΔz i nþ1 2 ¼ exp − Δz∇⊥ exp − δεdz exp − Δz∇2⊥ En E 4k 2 zn 4k ð23Þ Eq. (23) indicates that, propagating the field over a distance Δz consists of a vacuum propagation of the field over a distance Δz=2, an incrementing of the phase in accordance with nonlinear medium changes, and followed by a vacuum propagation of the resulting field over a distance Δz=2. In fact, after the first upgrading of the phase, the half steps of propagation can be combined into single propagation step. On the other hand, the hydrodynamic equation in the isobaric condition is expressed as [3] ∂ρ ðγ−1Þα þ v∇ρ ¼ − I ∂t c2s

ð24Þ

where ρ and v are perturbations in density and velocity, cs is the sound speed, and γ is the specific heat ratio. Based on Eqs. (23) and (24), the 4D computer code of the timedependent propagation of high power laser beams through the atmosphere is obtained by means of a discrete Fourier transform method, which is described in [3] in detail.

where ΓðÞ is the Euler Gamma function, and Γð; Þ is the incomplete Gamma function. N is the beam order, and a is the beam aperture. The changes of centroid position versus the laser beam power P and the time t are examined by using the 4D computer code of the time-dependent propagation of high power laser beams through the atmosphere. In the numerical examples, a thin lens with focal length zf is assumed to be just before the plane z¼ 0, and the calculation parameters are taken as γ ¼ 1:4, n0 ¼ 1:00035, cs ¼ 340 m=s, λ ¼ 10:6 μm, a ¼0.1 m, N ¼ 6, β ¼ −0:25 m−1 , zf ¼ 3 km, and the standard atmosphere density ρ0 ¼ 1:302461 kg=m3 . The three-dimensional (3D) intensity distribution Iðx′; y′; 0Þ and its contour lines at the source plane are given in Fig. 1(a) and (b), respectively. Fig. 1 shows that the centroid position is not located on the propagation z-axis, but is located on the x-axis. We can obtain the centroid position hx′i ¼ −21:36 mm 4 0 in Fig. 1. In vacuum, 3D intensity distribution Iðx; y; zÞ and its contour lines at the focal plane are shown in Fig. 2(a) and (b), respectively. Fig. 2 shows that Iðx; y; zÞ attains Gaussian shape, and the centroid position is located on the z-axis, which is in agreement with the result obtained in Section 3 [see Eqs. (6) and (12)]. The wind velocity is assumed to be zero in Figs.3–8. The contour lines of Iðx; y; zÞ at the focal plane for different values of time t are shown in Fig. 3(b)–(g), respectively, where the contour lines of Iðx′; y′; 0Þ at the source plane are also depicted in Fig. 3(a) for comparison. It can be seen that the thermal blooming results in a beam spreading and a shift of centroid position, and the effects of thermal blooming increase as the time t increases. On the other hand, the centroid position of laser beams is also defined as [4] R∞ R∞ R∞ R∞  −∞ xIðx; y; zÞ dx dy −∞ yIðx; y; zÞ dx dy R R−∞ hxi ¼ R−∞ ; y ¼ ð26Þ ∞ ∞ ∞ R∞ −∞ −∞ Iðx; y; zÞ dx dy −∞ −∞ Iðx; y; zÞ dx dy Based on Eq. (26), the centroid position can be obtained by using a discrete method. Fig.4 gives the change of the centroid position at the focal plane versus time t. Fig. 4 indicates that hxi increases with increasing t. Though we have hx′i ¼ −21:36 mm 40 at the source plane, the centroid position at the focal plane shifts along the x-axis from hxi ¼ 0 when t¼0 to hxi ¼ 5:35 mm o 0 when t¼1 s. The contour lines of Iðx; y; zÞ at the focal plane for different values of laser beam power P are shown in Fig. 5(a)–(e), respectively. Fig. 6 shows the change of the centroid position at the focal plane versus laser beam power P. It can be seen that both the beam spreading and the shift of centroid position decrease as the laser beam power P decreases. When P¼ 0.2 kW, the laser beam becomes a Gaussian beam, and its centroid position is located on z-axis, which is the same as in vacuum. Therefore, the effect of thermal blooming can be ignored when the laser beam power P is small enough. The contour lines of Iðx; y; zÞ at the focal plane for different values of the absorption coefficient α are shown in Fig. 7(a)–(f), respectively. Fig. 8 gives the change of the centroid position at the focal plane versus the absorption coefficient α. From Figs. 7 and 8 it can be seen that both the beam spreading and the shift of centroid position decrease as α decreases. When α ¼ 0:01  10−5 =m , the laser beam becomes a Gaussian beam, which is the same as in vacuum. Therefore, the effect of thermal blooming can be ignored when the absorption coefficient α is small enough.

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Fig. 11. (a) Contour lines of Iðx′; y′; 0Þ at the source plane, and (b)–(e) contour lines of Iðx; y; zÞ at the focal plane for different values of cross wind velocity vx along negative xaxis when t ¼0.12 s, P ¼60 kW and α ¼ 1:252  10−5 =m. (a), (b) vx=0m/s, (c) vx=  0.1m/s, (d) vx=  0.2m/s, (e) vx=  0.4m/s, (f) vx= 0.6m/s.

The effects of the cross wind on the centroid position are studied in Figs. 9–12. The contour lines of Iðx; y; zÞ at the focal plane for different values of the cross wind velocity vx along the positive x-axis (i.e., vx 40) are shown in Fig. 9(b)–(g), respectively, where the contour lines of Iðx′; y′; 0Þ at the source plane are also plotted in Fig. 9(a) for comparison. Fig. 10 gives the change of the centroid position at the focal plane versus vx. On the other hand, the contour lines of Iðx; y; zÞ and the change of the centroid position at the focal plane when the cross wind is along the negative x-axis case (i.e., vx o0) are shown in Figs. 11 and 12, respectively. Figs. 9 and 10 indicate that for the vx 40 case, there exists a minimum of hxi versus vx, where the centroid position is farthest away from the propagation z-axis. In addition, hxi reaches an asymptotic value when vx is large enough. From Figs. 11 and 12 it can be seen that for the vx o0 case, hxi increases monotonously as the absolute value of vx (i.e., jvx j) increases. Furthermore, for the

vx o0 case, hxi also reaches an asymptotic value when jvx j is large enough. It is noted that the centroid position is farther away from the propagation z-axis for the vx o 0 case as compared to the vx 40 case when jvx j is large enough, but the smaller jvx j is needed to reach the asymptotic value for the vx o 0 case as compared to the vx 40 case.

5. Physical explanations The physical reason why the centroid position changes on propagation is that the far-field centroid position is not located on the propagation z-axis due to the field phase distortion and the decentred intensity. Therefore, for laser beams with the spherical aberration and the decentred intensity, the centroid position will change on propagation.

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For no wind case, the effect of thermal blooming increases as the time, the laser beam power and the absorption coefficient increase, and the centroid position may shift along x-axis from hxi 40 (or hxi o 0) at the source plane to hxi o 0 (or hxi 4 0) at the focal plane. When the absolute value of cross wind velocity is large enough, hxi reaches an asymptotic value. The different behaviours of the centroid position will appear when the cross wind is along positive x-axis and negative x-axis (i. e., vx 4 0 and vx o 0). In particular, the centroid position may be farthest away from the propagation z-axis when both vx 40 (or vx o 0) and hxi 4 0 (or hxi o0) at the source plane.

Acknowledgements Fig. 12. Change of the centroid position at the focal plane versus cross wind velocity vx along the negative x-axis when t ¼0.12 s, P ¼60 kW and α ¼ 1:252  10−5 =m.

On the other hand, if a laser beam starts with the decentred intensity at the source plane, the thermal blooming will also be decentred. Thus, the decentred field phase distortion will appear due to thermal blooming when the laser beam propagates through the atmosphere. For example, for a laser beam shown in Fig. 1, the effect of thermal blooming is largest in the position of intensity maximum, i.e., xmax ¼ 87:5 mm 40, and at this place the refractive index of atmosphere reduces heavily due to thermal blooming. Therefore, for no wind case, the position of intensity maximum will shift along x-axis from xmax 4 0 at the source plane to xmax o 0 at the focal plane, which results in a shifting of centroid position in the same direction. When the laser beam power P is small enough or the absorption coefficient α is small enough, the field phase distortion due to thermal blooming can be ignored, and the centroid position is located on the propagation z-axis at the focal plane. Additionally, the cross wind causes the decentred field phase distortion, and reduces thermal blooming. The decentred intensity distribution at the source plane results in the different behaviours of the centroid position when the cross wind is along the positive x-axis and the negative x-axis. 6. Conclusions In this paper, the effects of atmospheric turbulence, initial field amplitude, optical system and thermal blooming on the centroid position hxi of laser beams propagating through the atmosphere have been studied in detail. It has been shown that hxi is independent of turbulence under the average over the ensemble of the turbulent medium. However, hxi depends on the centroid positions at the source plane and in the far-field, and elements A and B of ray-transfer-matrix. For laser beams without the field phase distortion, the far-field centroid position is always zero, which is independent of the intensity distribution at the source plane. For laser beams with the centred intensity, the far-field centroid position is zero when the field phase function is even function (e.g., spherical aberration), but this situation reverses if the field phase function is odd function (e.g., tilt aberration). For laser beams with both field phase distortion and decentred intensity, the far-field centroid position is not zero in general. The analytical expressions for the far-field centroid position of laser beams with spherical and tilt aberrations have been derived. The centroid position hxi of laser beams with the decentred intensity will change on propagation due to thermal blooming.

Xiaoling Ji and Xinhong Jia acknowledge the support by the National Natural Science Foundation of China (NSFC) under grant 61178070, and by the financial support from the Construction Plan for Scientific Research Innovation Teams of Universities in Sichuan Province under grant 12TD008.

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