Y. Huang and B. Zhang
Vol. 30, No. 11 / November 2013 / J. Opt. Soc. Am. A
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Turbulence distance for laser beams propagating through non-Kolmogorov turbulence Yongping Huang1,2 and Bin Zhang1,* 2
1 College of Electronics Information, Sichuan University, Chengdu 610064, China Computational Physics Key Laboratory of Sichuan Province, Physics Research Institute, Yibin University, Yibin 644007, China *Corresponding author: [email protected]
Received June 19, 2013; revised October 6, 2013; accepted October 7, 2013; posted October 7, 2013 (Doc. ID 192562); published October 24, 2013 Based on the second-order moments and the non-Kolmogorov turbulence spectrum, the general analytical expression for the turbulence distance of laser beams propagating through non-Kolmogorov turbulence is derived, which depends on the non-Kolmogorov turbulence parameters including the generalized exponent parameter α, inner scale l0 , and outer scale L0 and the initial second-order moments of the beams at the plane of z 0. Taking the partially coherent Hermite–Gaussian linear array (PCHGLA) beam as an illustrative example, the effects of nonKolmogorov turbulence and array parameters on the turbulence distance are discussed in detail. The results show that the turbulence distance zMx α of PCHGLA beams through non-Kolmogorov turbulence first decreases to a dip and then increases with increasing α, and the value of zMx α increases with increasing beam number and beam order and decreasing coherence parameter, meaning less influence of non-Kolmogorov turbulence on partially coherent array beams than that of fully coherent array beams and a single partially coherent beam. However, the value of zMx α for PCHGLA beams first increases nonmonotonically with the increasing of the relative beam separation x00 for x00 ≤ 1 and increases monotonically as x00 increases for x00 > 1. Moreover, the variation behavior of the turbulence distance with the generalized exponent parameter, inner scale, and outer scale of the turbulence and the beam number is similar, but different with the relative beam separation for coherent and incoherent combination cases. © 2013 Optical Society of America OCIS codes: (010.1330) Atmospheric turbulence; (140.3290) Laser arrays; (260.1960) Diffraction theory. http://dx.doi.org/10.1364/JOSAA.30.002339
1. INTRODUCTION Laser propagation through atmospheric turbulence is of great importance in some practical applications related to laser atmospheric engineering, but the effects of atmospheric turbulence can seriously degrade the performance of laser beams propagating in atmosphere [1–7]. However, there exists a range within which the influence of the turbulence on laser beams is small enough and can be neglected. Thus, the range of turbulence-negligible propagation is called turbulence distance [8–10]. It was introduced as characteristic distance zT for partially coherent beams considering the effect of the turbulence only on the beam width or the angular spread [8,9]. However, the variation of the M 2 factor in turbulence can comprehensively characterize the effects of the turbulence on the propagation characteristics of laser beams due to the invariance of the M 2 factor in free space. Recently, Dan introduced the range of turbulence-negligible propagation according to the definition of M 2 factor in turbulence, i.e., the turbulence distance zM defined by using relative M 2 factor [10]. On the other hand, array beams are widely used in highpower systems, inertial confinement fusion, high-energy weapons, and free space optics communications, etc., [11–20]. So far, the propagation properties of a variety of linear, rectangular, and radial laser arrays propagating in free space and turbulence have been investigated [13–20], e.g., Ai and Dan [20] studied the turbulence distance of linear Gaussian Schell model (GSM) array beams, whereas the final analytical expression of the turbulence distance zM has not yet been given. 1084-7529/13/112339-08$15.00/0
Moreover, experiments and theory have indicated that the Kolmogorov model is sometimes incomplete to describe practical turbulent atmosphere [21–24]. To our knowledge, the turbulence distance of laser beams propagating through non-Kolmogorov turbulence has not yet been reported. The main purpose of this paper is to investigate the turbulence distance for laser beams passing through nonKolmogorov turbulence. The general analytical expression of turbulence distance is derived in Section 2, which depends on the non-Kolmogorov turbulence parameters including the generalized exponent parameter α, inner scale l0 , and outer scale L0 of the turbulence and the initial second-order moments of laser beams at the plane of z 0. Taking the partially coherent Hermite–Gaussian linear array (PCHGLA) beam as an illustrative example, the turbulence distance of the PCHGLA beams passing through non-Kolmogorov turbulence is studied by theoretical analysis and numerical simulation in Sections 3 and 4, where the influence of non-Kolmogorov turbulence and array parameters is stressed, and both coherent and incoherent combination cases are considered. In Section 5, the main results obtained in this paper are summarized.
2. EXPRESSION OF TURBULENCE DISTANCE FOR LASER BEAMS IN NON-KOLMOGOROV TURBULENCE According to Refs. [3,4,24], the second moments of partially coherent beams propagating through non-Kolmogorov turbulence along the x direction can be expressed as © 2013 Optical Society of America
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hx2 i hx2 i0 hθ2x i0 z2 Tαz3 ;
(1)
hθ2x i hθ2x i0 3Tαz;
(2)
hxθx i hθ2x i0 z 3Tαz2 ∕2;
(3)
where the angle brackets with the subscript 0 indicate the second moments at the plane z 0, and 2π 2 Tα 3
Z
∞ 0
0 ≤ κ < ∞;
3 < α < 4;
(5)
where α is the generalized exponent parameter and C~ 2n is a generalized refractive-index structure parameter with units m3−α . κ0 2π∕L0 (L0 , outer scale of the atmospheric turbulence), κm cα∕l0 (l0 , inner scale of the atmospheric turbulence), cα 2π∕3Γ5 − α∕2Aα1∕α−5 (Γx, Gamma function), and Aα Γα − 1 cosαπ∕2∕4π 2 . For the case of α 11∕3, L0 ∞ and l0 0, A11∕3 0.033, C~ 2n C 2n , Φn κ; α reduces to the case of the conventional Kolmogorov spectrum. Substituting Eq. (5) into Eq. (4), the turbulence quantity Tα is expressed as Z∞ 2 Tα π 2 Φn κκ3 dκ 3 0 2 Aαπ 2 C~ 2n 8π c2 α cα∕l0 2−α 2 α − 2 2 3α − 2 l0 L0 2 2 4−α 2 2 4π l0 α 4π l 2π × exp 2 Γ 2 − ; 2 02 − 2 : (6) 2 c αL0 L0 c α L20 Equation (6) indicates that Tα depends on the generalized exponent parameter α, inner scale l0 , and outer scale L0 of the turbulence. Based on the definition of second moments, the M 2 factor can be expressed as [25] q M 2x z 2k hx2 ihθ2x i − hxθx i2 :
(7)
Substituting Eqs. (1)–(3) into Eq. (7), we can obtain the analytical formula for the M 2 factor of beams propagating through non-Kolmogorov turbulence, i.e., M 2x α; z k4hx2 i0 hθ2x i0 12hx2 i0 Tαz 4hθ2x i0 Tαz3 3T 2 αz4 1∕2 :
(8)
For the sake of convenience for analysis, further introducing the relative M 2 factor defined as M 2xr z; α ≡ M 2x z; α∕ M 2x z 0 [10,20], we can readily obtain the analytical expression from Eq. (8), i.e.,
(9)
Substituting Eq. (9) into Eq. (10) yields a quartic equation, i.e.,
(4)
is the turbulence quantity denoting the effect of nonKolmogorov turbulence [3]. In Eq. (4), Φn κ; α is the spatial power spectrum of the refractive-index fluctuations of the non-Kolmogorov turbulent atmosphere, and can be expressed as [23]
s 3Tαz Tαz3 3T 2 αz4 1 : 2 2 hθx i0 hx i0 4hx2 i0 hθ2x i0
Thus, the turbulence distance zMx α of laser beams in the x direction was defined as [20] p (10) M 2xr zMx α 2:
z4Mx α
Φn κ; ακ3 dκ
exp−κ2 ∕κ2m Φn κ;α AαC~ 2n ; κ2 κ20 α∕2
M 2xr α; z
4hθ2x i0 z3Mx α 4hx2 i0 zMx α 4hx2 i0 hθ2x i0 − 0: (11) Tα 3Tα 3T 2 α
Among the four solutions of the quartic equation [Eq. (11)], only one is real and positive. After tedious calculation, the turbulence distance of laser beams through turbulence is given by 1∕2 hθ2 i 1 4hθ2x i20 Q zMx α − x 0 3Tα 2 9T 2 α 8 91∕2 > > < 3 2 2 2 2 2 16hθx i0 216hx i0 T α= 1 8hθx i0 h i − Q ; (12) 2 1∕2 > 4hθ2 i2 2> :9T α ; 27T 3 α 2x 0 Q 9T α
where Q−
p 32 3 2hx2 i0 hθ2x i0 P ; −p 3 2 3T αP 54
217 hx2 i30 hθ2x i30 256S 2 P 16S T 6 α S
27hx2 i20 4hθ2x i30 hx2 i0 − : T 4 α T 2 α
(13) 1 13 2
;
(14)
(15)
Equation (12) shows the main result of this paper, which indicates that the turbulence distance of laser beams through turbulence depends on the beam second moments at plane z 0, namely the initial beam parameters and the non-Kolmogorov turbulent parameters such as the generalized exponent parameter α, generalized refractive-index structure parameter C~ 2n , outer scale L0 , and inner scale l0 . Apparently, from Eq. (12), it is clear that the turbulence distance zMx α decreases with increasing turbulence quantity Tα, indicating that the effect of the turbulence on the propagation characteristics of laser beams is more obvious for the stronger turbulence. According to Eq. (12), we can easily obtain the turbulence distance zMx α only if the initial second moments at the plane z 0 of any kind of laser beams and the turbulence quantity Tα are given. It is worth noting that, for the case of α 11∕3, zMx 11∕3 can be readily revealed as the turbulence distance of the beam in the Kolmogorov turbulence.
3. TURBULENCE DISTANCE OF PARTIALLY COHERENT HERMITE–GAUSSIAN LINEAR ARRAY BEAMS PROPAGATING THROUGH NON-KOLMOGOROV TURBULENCE In Fig. 1, we consider the one-dimensional linear array consisting of N individual off-axis, partially coherent Hermite Gaussian (PCHG) beams propagating through atmospheric turbulence along the z axis in the half-space z > 0.
Y. Huang and B. Zhang
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0
Letting
x x0
Fig. 1. Schematic diagram of the one-dimensional PCHGLA beams.
Let us assume that the individual off-axis PCHG beams are coherently combined. The cross-spectral density function of PCHGLA beams at the plane of z 0 is given by the expression [17]
p x01 x02 ∕2 − i jx0 ∕2;
(19)
q x02 − x01 i − jx0 ;
(20)
the intensity of PCHGLA beams is written as
W x01 ; x02 ; 0
p 0 p 0 2x2 − jx0 2x1 − ix0 Hm Hm w0 w0 i−N−1∕2 j−N−1∕2 x0 − ix0 2 x02 − jx0 2 × exp − 1 w20 x0 − x02 − i − jx0 2 × exp − 1 ; (16) 2σ 2 N−1∕2 X
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Ix;z
N−1∕2 X
k 2πz
N−1 N−1 ZZ X 2 2 X N−1 i−N−1 2 j− 2
Hm
p p 2p−q∕2 2p q∕2 Hm w0 w0
2 2p2 q2 ∕2 q ikx exp − exp q−i−jx 0 z w20 2σ 20 ik (21) ×exp − q −i −jx0 2pi jx0 dpdq: 2z
×exp −
Thus, the second moment is given by [26] PN−1∕2 PN−1∕2 R∞ 2 1 2 02 2 x Ix;zdx w0 i−N−1∕2 j−N−1∕2 aij Lm O−2Lm Ox0 ij Lm O −∞ hx i R ∞ PN−1∕2 PN−1∕2 4 i−N−1∕2 j−N−1∕2 aij Lm O −∞ Ix;zdx PN−1∕2 PN−1∕2 1 1 2 1 −2 −4 −2 −2 i−N−1∕2 j−N−1∕2 aij Lm O−2Lm OLm Oβ −aij OLm O−4Lm O4Lm OLm Oβ 2Lm Oβ −4Lm Oβ 2 z ; P P N−1∕2 N−1∕2 k2 w20 i−N−1∕2 j−N−1∕2 aij Lm O 2
(22) where σ is the spatial correlation length of individual off-axis PCHG beam sources, w0 denotes the waist width of the Gaussian beam, H m · represents the Hermite polynomials, and x01 , x02 are the one-dimensional position of two arbitrary points at the plane of z 0. If Tα 0 in Eq. (1), Eq. (1) becomes the spatial second moments of the PCHGLA beams through free space, namely hx2 i hx2 i0 hθ2x i0 z2 :
where O i − j2 x02 0;
x02 0 2 −2 aij exp − i − j β 1 : 2
PN−1∕2 hθ2x i0
Lm denotes the mth order Laguerre polynomial, and L1 m and are the first and second derivatives of Lm , respectively. For the sake of convenience, we introduce x00 x0 ∕w0 , indicating the relative beam separation, and β σ∕w0 , denoting the beam coherence parameter [20]. On comparing Eq. (22) with Eq. (17), the hx2 i0 and hθ2x i0 of PCHGLA beams at the plane of z 0 are expressed as
PN−1∕2 −
PN−1∕2
i−N−1∕2
L2 m
PN−1∕2
1 02 j−N−1∕2 aij Lm O − 2Lm O x0 i PN−1∕2 PN−1∕2 4 i−N−1∕2 j−N−1∕2 aij Lm O
PN−1∕2
1 −2 j−N−1∕2 aij Lm O − 2Lm O Lm Oβ P P N−1∕2 k2 w20 N−1∕2 i−N−1∕2 j−N−1∕2 aij Lm O
i−N−1∕2
w20
i−N−1∕2
PN−1∕2
(24)
(17)
Based on the Huygens–Fresnel principle, the intensity of PCHGLA beams propagating in free space is expressed as [17] ZZ k W x01 ; x02 ; 0 Ix; z 2πz ik 02 0 0 x1 − x02 − 2x − x x dx01 dx02 : (18) × exp 1 2 2 2z
hx2 i0
(23)
−4 2L Oβ−2 − m j−N−1∕2 aij OLm Oβ P P N−1∕2 a k2 w20 N−1∕2 i−N−1∕2 j−N−1∕2 ij Lm O
PN−1∕2 −
;
1 j−N−1∕2 aij OLm O − 4Lm O P P N−1∕2 k2 w20 N−1∕2 i−N−1∕2 j−N−1∕2 aij Lm O
i−N−1∕2
−2 4L1 m Oβ
PN−1∕2
j2 Lm O
:
(25)
4L2 m O
(26)
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For the case of the PCHGLA beams with incoherent combination, the second moments of the PCHGLA beams are written as [17] hx2 i00
w20 N 2 − 1 02 x0 ; 2m 1 3 4
(27)
λ2 2m 1 β−2 : 4π 2 w20
(28)
hθ2x i00
Substituting Eqs. (25)–(28) into Eq. (12), we can obtain the analytical formulas for the turbulence distance zMx α of the PCHGLA beams propagating through non-Kolmogorov turbulence for coherent combination and incoherent combination, respectively. It can be shown that zMx α of PCHGLA beams depends on the beam parameters such as beam number N and relative beam separation x00 , beam order m, coherence parameter β, waist size w0 , wavelength λ, and the turbulent parameters including the generalized exponent parameter α, inner scale l0 , and outer scale L0 of the turbulence. For the case of m 0, we can readily obtain the turbulence distance of the GSM array beams for coherent combination and incoherent combination in non-Kolmogorov turbulence, respectively. Furthermore, when α 11∕3 and m 0, the
corresponding expression represents the turbulence distance of GSM array beams in the Kolmogorov turbulence, namely, which reduces to the solution of the Eq. (20) in [20]. For the case of β → ∞, the corresponding expressions of the fully coherent Hermite–Gaussian array beams for coherent and incoherent combination cases can be easily obtained. When N 1 and x00 0, Eqs. (25)–(28) correspond to the initial second moments of the single PCHG beam on the plane z 0, and together with the Eq. (12), the expression of the turbulence distance for a single PCHG beam in non-Kolmogorov turbulence can be derived.
4. NUMERICAL CALCULATION RESULTS AND ANALYSIS In this section, numerical calculations for the turbulence distance of PCHGLA beams passing through non-Kolmogorov turbulence are carried out. Some typical examples of results are compiled in Figs. 2–7. Figures 2(a)–2(c) plot the turbulence distance zMx α of PCHGLA beams in non-Kolmogorov turbulence versus the generalized exponent parameter α for different values of (a) the relative beam separation x00 and the beam number N, (b) the beam order m, and (c) the coherence parameter β, where the calculation parameters are C~ 2n 10−13 m3−α ,
Fig. 2. Turbulence distance zMx α of PCHGLA beams in non-Kolmogorov turbulence versus the generalized exponent parameter α for different values of (a) the relative beam separation x00 and the beam number N, (b) the beam order m, and (c) the coherence parameter β.
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Fig. 3. zMx α of PCHGLA beams in non-Kolmogorov turbulence versus the relative beam separation x00 for different values of (a) beam number N and (b) the generalized exponent parameter α.
w0 0.05 m, λ 850 nm, m 5, L0 100 m, l0 0.005 m, β 1, N 9, and x00 2. Figures 2(a)–2(c) show that the zMx α of PCHGLA beams through non-Kolmogorov turbulence first decrease to a dip and then increase with increasing α, and the physical reason is that Eq. (12) implies zMx α decreases with increasing turbulence quantity Tα, whereas Tα increases toward its maximum and then decreases with
Fig. 4. zMx α of PCHGLA beams in non-Kolmogorov turbulence for the coherent combination and incoherent combination cases versus the generalized exponent parameter α.
increasing α as shown in Fig. 1 of [27], and the value of zMx α increases with the increasing of beam number N and beam order m and the decreasing of coherence parameter β. Figures 2(b) and 2(c) indicate that the variation behavior of the zMx α of PCHGLA beams versus α is similar to those of GSM linear array beams, fully coherent HG linear array beams, and a single PCHG beam. Moreover, Fig. 2(b) implies that the zMx α of PCHGLA beams is larger than that of GSM linear array beams with the same array parameters and that of a single PCHG beam with the same beam order, indicating that the array beam is less influenced by non-Kolmogorov turbulence than a single beam under the same conditions. When α 11∕3, the values of turbulence distance of PCHGLA beams, GSM linear array beams, and a single PCHG beam through Kolmogorov turbulence are 5.83, 5.35, and 3.65 km, respectively. Obviously, the turbulence distance of PCHGLA beams is larger than those of both GSM linear array beams and a single PCHG beam in Kolmogorov turbulence. Figure 2(c) shows that the value of zMx α for the PCHGLA beams is larger than that of GSM linear array beams and fully coherent Hermite–Gaussian linear array beams, revealing that partially coherent array beams are less sensitive to non-Kolmogorov turbulence than fully coherent array beams under the same conditions. For the case of α 11∕3, the values of turbulence distance of PCHGLA beams, GSM linear array beams, and fully coherent Hermite–Gaussian linear array beams through
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Fig. 5. zMx α of PCHGLA beams in non-Kolmogorov turbulence for the coherent combination and incoherent combination cases versus (a) inner scale and (b) outer scale.
Kolmogorov turbulence are 4.91, 4.80, and 4.18 km, respectively, indicating that the turbulence distance of PCHGLA beams is larger than those of both GSM linear array beams and fully coherent Hermite–Gaussian linear array beams in Kolmogorov turbulence. Therefore, we can determine that the turbulence distance for PCHGLA beams is larger than those of both fully coherent linear array beams and a single partially coherent beam propagating through Kolmogorov turbulence, which agrees well with the results reported in [20]. Figure 3 plots the zMx α of PCHGLA beams in nonKolmogorov turbulence versus the relative beam separation x00 for different values of (a) beam number N and (b) the generalized exponent parameter α, where (a) α 3.11, (b) N 9, and the other calculation parameters are the same as those in Fig. 2. From Figs. 3(a) and 3(b), we can see that the zMx α first varies nonmonotonically with the increasing x00 for x00 ≤ 1 and increases monotonically with x00 for x00 > 1. The physical reason is that the adjacent beams overlap for x00 ≤ 1 (namely, x0 ≤ w0 ), resulting in the significant influence of the beams
on each other. On the other hand, for a given x00 , the value of zMx α noticeably increases with increasing N for x00 > 1, as shown in Fig. 3(a). Moreover, from Fig. 3(b), it is clear that the value of zMx α is different for different values of α, and exhibits more prominent difference for x00 > 1, indicating that the turbulence distance can be evidently influenced by generalized exponent parameter α of non-Kolmogorov turbulence for x00 > 1. Figures 4 and 5 give zMx α of PCHGLA beams in nonKolmogorov turbulence for coherent and incoherent combination cases versus the generalized exponent parameter α, inner scale l0 , and outer scale L0 . The calculation parameters are C~ 2n 10−13 m3−α , β 0.3; the other calculation parameters are the same as those in Fig. 2. The solid curve indicates the case of coherent combination and the dashed curve corresponds to the case of incoherent combination. From Figs. 4 and 5, it can be seen that the variation behavior of the zMx α of PCHGLA beams is similar for both coherent and incoherent combination cases, i.e., zMx α first decreases and reaches to its minimum, and then increases with increasing α, and increases monotonically with increasing l0 and
Fig. 6. zMx α of PCHGLA beams in non-Kolmogorov turbulence for the coherent combination and incoherent combination cases versus beam number N.
Fig. 7. zMx α of PCHGLA beams in non-Kolmogorov turbulence for the coherent combination and incoherent combination cases versus the relative beam separation x00 .
Y. Huang and B. Zhang
decreases with increasing L0 . The value of zMx α of PCHGLA beams for incoherent combination is larger than that of the coherent combination case, implying the lesser influence of non-Kolmogorov turbulence on the partially coherent array beams for the incoherent combination case. Figure 6 plots the zMx α of PCHGLA beams in nonKolmogorov turbulence for coherent and incoherent combination cases versus the beam number N, where x00 2, α 3.3, C~ 2n 10−14 m3−α , and β 0.3; the other calculation parameters are the same as those of Fig. 2. The solid curve indicates the case of incoherent combination and the dots correspond to the case of coherent combination. As can be seen from Fig. 6, zMx α for both coherent and incoherent combination cases increases with increasing N, but the difference values of zMx α between both cases becomes smaller and smaller, and even almost disappears as N increases, indicating that the PCHGLA beam with enough large beam number N through non-Kolmogorov turbulence is less influenced by different combination cases. Figure 7 gives the zMx α of PCHGLA beams in nonKolmogorov turbulence for coherent and incoherent combination cases versus the relative beam separation x00 , where N 9. The other calculation parameters are the same as those of Fig. 6, and the solid and dashed curves indicate the cases of the coherent combination and incoherent combination, respectively. It can be shown from Fig. 7 that zMx α of PCHGLA beams for the incoherent combination increases monotonically with increasing x00 , whereas it is not the case for the coherent combination. The difference between both cases is larger for x00 ≤ 1, whereas it becomes smaller with increasing x00 for x00 > 1, and turbulence distance is larger for the larger x00 . These results imply less influence of different combination cases on PCHGLA beams with enough large x00 in non-Kolmogorov turbulence.
5. CONCLUDING REMARKS In this paper, based on the second-order moments and the non-Kolmogorov spectrum, the general analytical expression for the turbulence distance zMx α of laser beams propagating through non-Kolmogorov turbulence is derived. It is shown that the turbulence distance depends on the non-Kolmogorov turbulence parameters, including the generalized exponent parameter α, inner scale l0 , and outer scale L0 of the turbulence and the initial second-order moments of the beams at the plane of z 0. According to the result, the turbulence distance of any kind of laser beams can be easily obtained only if the initial second moments at the plane z 0 and the turbulence quantity Tα are given. The zMx α of PCHGLA beams propagating through non-Kolmogorov turbulence has been discussed in detail, where the influence of non-Kolmogorov turbulent atmosphere and array beam parameters on zMx α have been stressed and both coherent and incoherent combination cases have also been considered. It can be shown that the turbulence distance zMx α of PCHGLA beams in non-Kolmogorov turbulence varies nonmonotonically with increasing α, and the value of zMx α increases with increasing beam number N, beam order m, and decreasing coherence parameter β, meaning less influence of non-Kolmogorov turbulence on partially coherent array beams than that of fully coherent array beams and a single partially coherent beam. Furthermore, it varies nonmonotonically oscillatory with
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increasing relative beam separation x00 in the region x00 ≤ 1, and then increases monotonically with x00 in the region x00 > 1. For coherent and incoherent combination cases, the variation behavior of zMx α of PCHGLA beams in nonKolmogorov turbulence with α, l0 , L0 , and N is similar, but different with x00 in the region x00 ≤ 1. The difference of turbulence distance between coherent and incoherent combination cases is smaller for the PCHGLA with enough large N and x00 . According to the results obtained in the paper, the turbulence distance of GSM linear array beams, fully coherent HG array beams, and a single PCHG beam in non-Kolmogorov turbulence can be investigated easily as special cases of PCHGLA beams.
ACKNOWLEDGMENTS This work is supported by the National Natural Science Foundation of China under Grant No. 61275203, the National Natural Science Foundation of China and Civil Aviation Administration of China under Grant No. 61079023, the Science Foundation of Sichuan Provincial Education Department under Grant No. 12ZA203, the Program for Bureau of Yibin City Science and Technology under Grant No. 2013SF020, the Program for Excellent Youth Talents in Sichuan University under Grant No. 2011-2-B17, and the Program for Innovation Team of Sichuan Provincial Education Department under Grant No. 13TD0048.
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15. X. Du and D. Zhao, “Propagation characteristics of stochastic electromagnetic array beams,” Appl. Phys. B 93, 901–905 (2008). 16. H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Scintillations of laser array beams,” Appl. Phys. B 91, 265–271 (2008). 17. X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite–Gaussian array beams,” J. Opt. A 11, 105705 (2009). 18. X. Ji and Z. Pu, “Effective Rayleigh range of Gaussian array beams propagating through atmospheric turbulence,” Opt. Commun. 283, 3884–3890 (2010). 19. Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467–475 (2007). 20. Y. L. Ai and Y. Q. Dan, “Range of turbulence-negligible propagation of Gaussian Schell-model array beams,” Opt. Commun. 284, 3216–3220 (2011). 21. M. S. Belen’kii, S. J. Karis, J. M. Brown II, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
Y. Huang and B. Zhang 22. M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown II, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect on non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999). 23. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007). 24. Y. P. Huang, Z. H. Gao, and B. Zhang, “The Rayleigh range of elegant Hermite–Gaussian beams in non-Kolmogorov turbulence,” Opt. Lasers Technol. 50, 125–129 (2013). 25. J. Serna, R. Martínez-Herrero, and P. M. Mejías, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991). 26. H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, S1027– S1049 (1992). 27. Y. P. Huang, G. P. Zhao, D. He, and Z. C. Duan, “Spreading and M2-factor of elegant Hermite-Gaussian beams through nonKolmogorov turbulence,” J. Mod. Opt. 58, 912–917 (2011).
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Turbulence distance for laser beams propagating through non-Kolmogorov turbulence Yongping Huang1,2 and Bin Zhang1,* 2
1 College of Electronics Information, Sichuan University, Chengdu 610064, China Computational Physics Key Laboratory of Sichuan Province, Physics Research Institute, Yibin University, Yibin 644007, China *Corresponding author: [email protected]
Received June 19, 2013; revised October 6, 2013; accepted October 7, 2013; posted October 7, 2013 (Doc. ID 192562); published October 24, 2013 Based on the second-order moments and the non-Kolmogorov turbulence spectrum, the general analytical expression for the turbulence distance of laser beams propagating through non-Kolmogorov turbulence is derived, which depends on the non-Kolmogorov turbulence parameters including the generalized exponent parameter α, inner scale l0 , and outer scale L0 and the initial second-order moments of the beams at the plane of z 0. Taking the partially coherent Hermite–Gaussian linear array (PCHGLA) beam as an illustrative example, the effects of nonKolmogorov turbulence and array parameters on the turbulence distance are discussed in detail. The results show that the turbulence distance zMx α of PCHGLA beams through non-Kolmogorov turbulence first decreases to a dip and then increases with increasing α, and the value of zMx α increases with increasing beam number and beam order and decreasing coherence parameter, meaning less influence of non-Kolmogorov turbulence on partially coherent array beams than that of fully coherent array beams and a single partially coherent beam. However, the value of zMx α for PCHGLA beams first increases nonmonotonically with the increasing of the relative beam separation x00 for x00 ≤ 1 and increases monotonically as x00 increases for x00 > 1. Moreover, the variation behavior of the turbulence distance with the generalized exponent parameter, inner scale, and outer scale of the turbulence and the beam number is similar, but different with the relative beam separation for coherent and incoherent combination cases. © 2013 Optical Society of America OCIS codes: (010.1330) Atmospheric turbulence; (140.3290) Laser arrays; (260.1960) Diffraction theory. http://dx.doi.org/10.1364/JOSAA.30.002339
1. INTRODUCTION Laser propagation through atmospheric turbulence is of great importance in some practical applications related to laser atmospheric engineering, but the effects of atmospheric turbulence can seriously degrade the performance of laser beams propagating in atmosphere [1–7]. However, there exists a range within which the influence of the turbulence on laser beams is small enough and can be neglected. Thus, the range of turbulence-negligible propagation is called turbulence distance [8–10]. It was introduced as characteristic distance zT for partially coherent beams considering the effect of the turbulence only on the beam width or the angular spread [8,9]. However, the variation of the M 2 factor in turbulence can comprehensively characterize the effects of the turbulence on the propagation characteristics of laser beams due to the invariance of the M 2 factor in free space. Recently, Dan introduced the range of turbulence-negligible propagation according to the definition of M 2 factor in turbulence, i.e., the turbulence distance zM defined by using relative M 2 factor [10]. On the other hand, array beams are widely used in highpower systems, inertial confinement fusion, high-energy weapons, and free space optics communications, etc., [11–20]. So far, the propagation properties of a variety of linear, rectangular, and radial laser arrays propagating in free space and turbulence have been investigated [13–20], e.g., Ai and Dan [20] studied the turbulence distance of linear Gaussian Schell model (GSM) array beams, whereas the final analytical expression of the turbulence distance zM has not yet been given. 1084-7529/13/112339-08$15.00/0
Moreover, experiments and theory have indicated that the Kolmogorov model is sometimes incomplete to describe practical turbulent atmosphere [21–24]. To our knowledge, the turbulence distance of laser beams propagating through non-Kolmogorov turbulence has not yet been reported. The main purpose of this paper is to investigate the turbulence distance for laser beams passing through nonKolmogorov turbulence. The general analytical expression of turbulence distance is derived in Section 2, which depends on the non-Kolmogorov turbulence parameters including the generalized exponent parameter α, inner scale l0 , and outer scale L0 of the turbulence and the initial second-order moments of laser beams at the plane of z 0. Taking the partially coherent Hermite–Gaussian linear array (PCHGLA) beam as an illustrative example, the turbulence distance of the PCHGLA beams passing through non-Kolmogorov turbulence is studied by theoretical analysis and numerical simulation in Sections 3 and 4, where the influence of non-Kolmogorov turbulence and array parameters is stressed, and both coherent and incoherent combination cases are considered. In Section 5, the main results obtained in this paper are summarized.
2. EXPRESSION OF TURBULENCE DISTANCE FOR LASER BEAMS IN NON-KOLMOGOROV TURBULENCE According to Refs. [3,4,24], the second moments of partially coherent beams propagating through non-Kolmogorov turbulence along the x direction can be expressed as © 2013 Optical Society of America
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Y. Huang and B. Zhang
hx2 i hx2 i0 hθ2x i0 z2 Tαz3 ;
(1)
hθ2x i hθ2x i0 3Tαz;
(2)
hxθx i hθ2x i0 z 3Tαz2 ∕2;
(3)
where the angle brackets with the subscript 0 indicate the second moments at the plane z 0, and 2π 2 Tα 3
Z
∞ 0
0 ≤ κ < ∞;
3 < α < 4;
(5)
where α is the generalized exponent parameter and C~ 2n is a generalized refractive-index structure parameter with units m3−α . κ0 2π∕L0 (L0 , outer scale of the atmospheric turbulence), κm cα∕l0 (l0 , inner scale of the atmospheric turbulence), cα 2π∕3Γ5 − α∕2Aα1∕α−5 (Γx, Gamma function), and Aα Γα − 1 cosαπ∕2∕4π 2 . For the case of α 11∕3, L0 ∞ and l0 0, A11∕3 0.033, C~ 2n C 2n , Φn κ; α reduces to the case of the conventional Kolmogorov spectrum. Substituting Eq. (5) into Eq. (4), the turbulence quantity Tα is expressed as Z∞ 2 Tα π 2 Φn κκ3 dκ 3 0 2 Aαπ 2 C~ 2n 8π c2 α cα∕l0 2−α 2 α − 2 2 3α − 2 l0 L0 2 2 4−α 2 2 4π l0 α 4π l 2π × exp 2 Γ 2 − ; 2 02 − 2 : (6) 2 c αL0 L0 c α L20 Equation (6) indicates that Tα depends on the generalized exponent parameter α, inner scale l0 , and outer scale L0 of the turbulence. Based on the definition of second moments, the M 2 factor can be expressed as [25] q M 2x z 2k hx2 ihθ2x i − hxθx i2 :
(7)
Substituting Eqs. (1)–(3) into Eq. (7), we can obtain the analytical formula for the M 2 factor of beams propagating through non-Kolmogorov turbulence, i.e., M 2x α; z k4hx2 i0 hθ2x i0 12hx2 i0 Tαz 4hθ2x i0 Tαz3 3T 2 αz4 1∕2 :
(8)
For the sake of convenience for analysis, further introducing the relative M 2 factor defined as M 2xr z; α ≡ M 2x z; α∕ M 2x z 0 [10,20], we can readily obtain the analytical expression from Eq. (8), i.e.,
(9)
Substituting Eq. (9) into Eq. (10) yields a quartic equation, i.e.,
(4)
is the turbulence quantity denoting the effect of nonKolmogorov turbulence [3]. In Eq. (4), Φn κ; α is the spatial power spectrum of the refractive-index fluctuations of the non-Kolmogorov turbulent atmosphere, and can be expressed as [23]
s 3Tαz Tαz3 3T 2 αz4 1 : 2 2 hθx i0 hx i0 4hx2 i0 hθ2x i0
Thus, the turbulence distance zMx α of laser beams in the x direction was defined as [20] p (10) M 2xr zMx α 2:
z4Mx α
Φn κ; ακ3 dκ
exp−κ2 ∕κ2m Φn κ;α AαC~ 2n ; κ2 κ20 α∕2
M 2xr α; z
4hθ2x i0 z3Mx α 4hx2 i0 zMx α 4hx2 i0 hθ2x i0 − 0: (11) Tα 3Tα 3T 2 α
Among the four solutions of the quartic equation [Eq. (11)], only one is real and positive. After tedious calculation, the turbulence distance of laser beams through turbulence is given by 1∕2 hθ2 i 1 4hθ2x i20 Q zMx α − x 0 3Tα 2 9T 2 α 8 91∕2 > > < 3 2 2 2 2 2 16hθx i0 216hx i0 T α= 1 8hθx i0 h i − Q ; (12) 2 1∕2 > 4hθ2 i2 2> :9T α ; 27T 3 α 2x 0 Q 9T α
where Q−
p 32 3 2hx2 i0 hθ2x i0 P ; −p 3 2 3T αP 54
217 hx2 i30 hθ2x i30 256S 2 P 16S T 6 α S
27hx2 i20 4hθ2x i30 hx2 i0 − : T 4 α T 2 α
(13) 1 13 2
;
(14)
(15)
Equation (12) shows the main result of this paper, which indicates that the turbulence distance of laser beams through turbulence depends on the beam second moments at plane z 0, namely the initial beam parameters and the non-Kolmogorov turbulent parameters such as the generalized exponent parameter α, generalized refractive-index structure parameter C~ 2n , outer scale L0 , and inner scale l0 . Apparently, from Eq. (12), it is clear that the turbulence distance zMx α decreases with increasing turbulence quantity Tα, indicating that the effect of the turbulence on the propagation characteristics of laser beams is more obvious for the stronger turbulence. According to Eq. (12), we can easily obtain the turbulence distance zMx α only if the initial second moments at the plane z 0 of any kind of laser beams and the turbulence quantity Tα are given. It is worth noting that, for the case of α 11∕3, zMx 11∕3 can be readily revealed as the turbulence distance of the beam in the Kolmogorov turbulence.
3. TURBULENCE DISTANCE OF PARTIALLY COHERENT HERMITE–GAUSSIAN LINEAR ARRAY BEAMS PROPAGATING THROUGH NON-KOLMOGOROV TURBULENCE In Fig. 1, we consider the one-dimensional linear array consisting of N individual off-axis, partially coherent Hermite Gaussian (PCHG) beams propagating through atmospheric turbulence along the z axis in the half-space z > 0.
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0
Letting
x x0
Fig. 1. Schematic diagram of the one-dimensional PCHGLA beams.
Let us assume that the individual off-axis PCHG beams are coherently combined. The cross-spectral density function of PCHGLA beams at the plane of z 0 is given by the expression [17]
p x01 x02 ∕2 − i jx0 ∕2;
(19)
q x02 − x01 i − jx0 ;
(20)
the intensity of PCHGLA beams is written as
W x01 ; x02 ; 0
p 0 p 0 2x2 − jx0 2x1 − ix0 Hm Hm w0 w0 i−N−1∕2 j−N−1∕2 x0 − ix0 2 x02 − jx0 2 × exp − 1 w20 x0 − x02 − i − jx0 2 × exp − 1 ; (16) 2σ 2 N−1∕2 X
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Ix;z
N−1∕2 X
k 2πz
N−1 N−1 ZZ X 2 2 X N−1 i−N−1 2 j− 2
Hm
p p 2p−q∕2 2p q∕2 Hm w0 w0
2 2p2 q2 ∕2 q ikx exp − exp q−i−jx 0 z w20 2σ 20 ik (21) ×exp − q −i −jx0 2pi jx0 dpdq: 2z
×exp −
Thus, the second moment is given by [26] PN−1∕2 PN−1∕2 R∞ 2 1 2 02 2 x Ix;zdx w0 i−N−1∕2 j−N−1∕2 aij Lm O−2Lm Ox0 ij Lm O −∞ hx i R ∞ PN−1∕2 PN−1∕2 4 i−N−1∕2 j−N−1∕2 aij Lm O −∞ Ix;zdx PN−1∕2 PN−1∕2 1 1 2 1 −2 −4 −2 −2 i−N−1∕2 j−N−1∕2 aij Lm O−2Lm OLm Oβ −aij OLm O−4Lm O4Lm OLm Oβ 2Lm Oβ −4Lm Oβ 2 z ; P P N−1∕2 N−1∕2 k2 w20 i−N−1∕2 j−N−1∕2 aij Lm O 2
(22) where σ is the spatial correlation length of individual off-axis PCHG beam sources, w0 denotes the waist width of the Gaussian beam, H m · represents the Hermite polynomials, and x01 , x02 are the one-dimensional position of two arbitrary points at the plane of z 0. If Tα 0 in Eq. (1), Eq. (1) becomes the spatial second moments of the PCHGLA beams through free space, namely hx2 i hx2 i0 hθ2x i0 z2 :
where O i − j2 x02 0;
x02 0 2 −2 aij exp − i − j β 1 : 2
PN−1∕2 hθ2x i0
Lm denotes the mth order Laguerre polynomial, and L1 m and are the first and second derivatives of Lm , respectively. For the sake of convenience, we introduce x00 x0 ∕w0 , indicating the relative beam separation, and β σ∕w0 , denoting the beam coherence parameter [20]. On comparing Eq. (22) with Eq. (17), the hx2 i0 and hθ2x i0 of PCHGLA beams at the plane of z 0 are expressed as
PN−1∕2 −
PN−1∕2
i−N−1∕2
L2 m
PN−1∕2
1 02 j−N−1∕2 aij Lm O − 2Lm O x0 i PN−1∕2 PN−1∕2 4 i−N−1∕2 j−N−1∕2 aij Lm O
PN−1∕2
1 −2 j−N−1∕2 aij Lm O − 2Lm O Lm Oβ P P N−1∕2 k2 w20 N−1∕2 i−N−1∕2 j−N−1∕2 aij Lm O
i−N−1∕2
w20
i−N−1∕2
PN−1∕2
(24)
(17)
Based on the Huygens–Fresnel principle, the intensity of PCHGLA beams propagating in free space is expressed as [17] ZZ k W x01 ; x02 ; 0 Ix; z 2πz ik 02 0 0 x1 − x02 − 2x − x x dx01 dx02 : (18) × exp 1 2 2 2z
hx2 i0
(23)
−4 2L Oβ−2 − m j−N−1∕2 aij OLm Oβ P P N−1∕2 a k2 w20 N−1∕2 i−N−1∕2 j−N−1∕2 ij Lm O
PN−1∕2 −
;
1 j−N−1∕2 aij OLm O − 4Lm O P P N−1∕2 k2 w20 N−1∕2 i−N−1∕2 j−N−1∕2 aij Lm O
i−N−1∕2
−2 4L1 m Oβ
PN−1∕2
j2 Lm O
:
(25)
4L2 m O
(26)
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For the case of the PCHGLA beams with incoherent combination, the second moments of the PCHGLA beams are written as [17] hx2 i00
w20 N 2 − 1 02 x0 ; 2m 1 3 4
(27)
λ2 2m 1 β−2 : 4π 2 w20
(28)
hθ2x i00
Substituting Eqs. (25)–(28) into Eq. (12), we can obtain the analytical formulas for the turbulence distance zMx α of the PCHGLA beams propagating through non-Kolmogorov turbulence for coherent combination and incoherent combination, respectively. It can be shown that zMx α of PCHGLA beams depends on the beam parameters such as beam number N and relative beam separation x00 , beam order m, coherence parameter β, waist size w0 , wavelength λ, and the turbulent parameters including the generalized exponent parameter α, inner scale l0 , and outer scale L0 of the turbulence. For the case of m 0, we can readily obtain the turbulence distance of the GSM array beams for coherent combination and incoherent combination in non-Kolmogorov turbulence, respectively. Furthermore, when α 11∕3 and m 0, the
corresponding expression represents the turbulence distance of GSM array beams in the Kolmogorov turbulence, namely, which reduces to the solution of the Eq. (20) in [20]. For the case of β → ∞, the corresponding expressions of the fully coherent Hermite–Gaussian array beams for coherent and incoherent combination cases can be easily obtained. When N 1 and x00 0, Eqs. (25)–(28) correspond to the initial second moments of the single PCHG beam on the plane z 0, and together with the Eq. (12), the expression of the turbulence distance for a single PCHG beam in non-Kolmogorov turbulence can be derived.
4. NUMERICAL CALCULATION RESULTS AND ANALYSIS In this section, numerical calculations for the turbulence distance of PCHGLA beams passing through non-Kolmogorov turbulence are carried out. Some typical examples of results are compiled in Figs. 2–7. Figures 2(a)–2(c) plot the turbulence distance zMx α of PCHGLA beams in non-Kolmogorov turbulence versus the generalized exponent parameter α for different values of (a) the relative beam separation x00 and the beam number N, (b) the beam order m, and (c) the coherence parameter β, where the calculation parameters are C~ 2n 10−13 m3−α ,
Fig. 2. Turbulence distance zMx α of PCHGLA beams in non-Kolmogorov turbulence versus the generalized exponent parameter α for different values of (a) the relative beam separation x00 and the beam number N, (b) the beam order m, and (c) the coherence parameter β.
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Fig. 3. zMx α of PCHGLA beams in non-Kolmogorov turbulence versus the relative beam separation x00 for different values of (a) beam number N and (b) the generalized exponent parameter α.
w0 0.05 m, λ 850 nm, m 5, L0 100 m, l0 0.005 m, β 1, N 9, and x00 2. Figures 2(a)–2(c) show that the zMx α of PCHGLA beams through non-Kolmogorov turbulence first decrease to a dip and then increase with increasing α, and the physical reason is that Eq. (12) implies zMx α decreases with increasing turbulence quantity Tα, whereas Tα increases toward its maximum and then decreases with
Fig. 4. zMx α of PCHGLA beams in non-Kolmogorov turbulence for the coherent combination and incoherent combination cases versus the generalized exponent parameter α.
increasing α as shown in Fig. 1 of [27], and the value of zMx α increases with the increasing of beam number N and beam order m and the decreasing of coherence parameter β. Figures 2(b) and 2(c) indicate that the variation behavior of the zMx α of PCHGLA beams versus α is similar to those of GSM linear array beams, fully coherent HG linear array beams, and a single PCHG beam. Moreover, Fig. 2(b) implies that the zMx α of PCHGLA beams is larger than that of GSM linear array beams with the same array parameters and that of a single PCHG beam with the same beam order, indicating that the array beam is less influenced by non-Kolmogorov turbulence than a single beam under the same conditions. When α 11∕3, the values of turbulence distance of PCHGLA beams, GSM linear array beams, and a single PCHG beam through Kolmogorov turbulence are 5.83, 5.35, and 3.65 km, respectively. Obviously, the turbulence distance of PCHGLA beams is larger than those of both GSM linear array beams and a single PCHG beam in Kolmogorov turbulence. Figure 2(c) shows that the value of zMx α for the PCHGLA beams is larger than that of GSM linear array beams and fully coherent Hermite–Gaussian linear array beams, revealing that partially coherent array beams are less sensitive to non-Kolmogorov turbulence than fully coherent array beams under the same conditions. For the case of α 11∕3, the values of turbulence distance of PCHGLA beams, GSM linear array beams, and fully coherent Hermite–Gaussian linear array beams through
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Fig. 5. zMx α of PCHGLA beams in non-Kolmogorov turbulence for the coherent combination and incoherent combination cases versus (a) inner scale and (b) outer scale.
Kolmogorov turbulence are 4.91, 4.80, and 4.18 km, respectively, indicating that the turbulence distance of PCHGLA beams is larger than those of both GSM linear array beams and fully coherent Hermite–Gaussian linear array beams in Kolmogorov turbulence. Therefore, we can determine that the turbulence distance for PCHGLA beams is larger than those of both fully coherent linear array beams and a single partially coherent beam propagating through Kolmogorov turbulence, which agrees well with the results reported in [20]. Figure 3 plots the zMx α of PCHGLA beams in nonKolmogorov turbulence versus the relative beam separation x00 for different values of (a) beam number N and (b) the generalized exponent parameter α, where (a) α 3.11, (b) N 9, and the other calculation parameters are the same as those in Fig. 2. From Figs. 3(a) and 3(b), we can see that the zMx α first varies nonmonotonically with the increasing x00 for x00 ≤ 1 and increases monotonically with x00 for x00 > 1. The physical reason is that the adjacent beams overlap for x00 ≤ 1 (namely, x0 ≤ w0 ), resulting in the significant influence of the beams
on each other. On the other hand, for a given x00 , the value of zMx α noticeably increases with increasing N for x00 > 1, as shown in Fig. 3(a). Moreover, from Fig. 3(b), it is clear that the value of zMx α is different for different values of α, and exhibits more prominent difference for x00 > 1, indicating that the turbulence distance can be evidently influenced by generalized exponent parameter α of non-Kolmogorov turbulence for x00 > 1. Figures 4 and 5 give zMx α of PCHGLA beams in nonKolmogorov turbulence for coherent and incoherent combination cases versus the generalized exponent parameter α, inner scale l0 , and outer scale L0 . The calculation parameters are C~ 2n 10−13 m3−α , β 0.3; the other calculation parameters are the same as those in Fig. 2. The solid curve indicates the case of coherent combination and the dashed curve corresponds to the case of incoherent combination. From Figs. 4 and 5, it can be seen that the variation behavior of the zMx α of PCHGLA beams is similar for both coherent and incoherent combination cases, i.e., zMx α first decreases and reaches to its minimum, and then increases with increasing α, and increases monotonically with increasing l0 and
Fig. 6. zMx α of PCHGLA beams in non-Kolmogorov turbulence for the coherent combination and incoherent combination cases versus beam number N.
Fig. 7. zMx α of PCHGLA beams in non-Kolmogorov turbulence for the coherent combination and incoherent combination cases versus the relative beam separation x00 .
Y. Huang and B. Zhang
decreases with increasing L0 . The value of zMx α of PCHGLA beams for incoherent combination is larger than that of the coherent combination case, implying the lesser influence of non-Kolmogorov turbulence on the partially coherent array beams for the incoherent combination case. Figure 6 plots the zMx α of PCHGLA beams in nonKolmogorov turbulence for coherent and incoherent combination cases versus the beam number N, where x00 2, α 3.3, C~ 2n 10−14 m3−α , and β 0.3; the other calculation parameters are the same as those of Fig. 2. The solid curve indicates the case of incoherent combination and the dots correspond to the case of coherent combination. As can be seen from Fig. 6, zMx α for both coherent and incoherent combination cases increases with increasing N, but the difference values of zMx α between both cases becomes smaller and smaller, and even almost disappears as N increases, indicating that the PCHGLA beam with enough large beam number N through non-Kolmogorov turbulence is less influenced by different combination cases. Figure 7 gives the zMx α of PCHGLA beams in nonKolmogorov turbulence for coherent and incoherent combination cases versus the relative beam separation x00 , where N 9. The other calculation parameters are the same as those of Fig. 6, and the solid and dashed curves indicate the cases of the coherent combination and incoherent combination, respectively. It can be shown from Fig. 7 that zMx α of PCHGLA beams for the incoherent combination increases monotonically with increasing x00 , whereas it is not the case for the coherent combination. The difference between both cases is larger for x00 ≤ 1, whereas it becomes smaller with increasing x00 for x00 > 1, and turbulence distance is larger for the larger x00 . These results imply less influence of different combination cases on PCHGLA beams with enough large x00 in non-Kolmogorov turbulence.
5. CONCLUDING REMARKS In this paper, based on the second-order moments and the non-Kolmogorov spectrum, the general analytical expression for the turbulence distance zMx α of laser beams propagating through non-Kolmogorov turbulence is derived. It is shown that the turbulence distance depends on the non-Kolmogorov turbulence parameters, including the generalized exponent parameter α, inner scale l0 , and outer scale L0 of the turbulence and the initial second-order moments of the beams at the plane of z 0. According to the result, the turbulence distance of any kind of laser beams can be easily obtained only if the initial second moments at the plane z 0 and the turbulence quantity Tα are given. The zMx α of PCHGLA beams propagating through non-Kolmogorov turbulence has been discussed in detail, where the influence of non-Kolmogorov turbulent atmosphere and array beam parameters on zMx α have been stressed and both coherent and incoherent combination cases have also been considered. It can be shown that the turbulence distance zMx α of PCHGLA beams in non-Kolmogorov turbulence varies nonmonotonically with increasing α, and the value of zMx α increases with increasing beam number N, beam order m, and decreasing coherence parameter β, meaning less influence of non-Kolmogorov turbulence on partially coherent array beams than that of fully coherent array beams and a single partially coherent beam. Furthermore, it varies nonmonotonically oscillatory with
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increasing relative beam separation x00 in the region x00 ≤ 1, and then increases monotonically with x00 in the region x00 > 1. For coherent and incoherent combination cases, the variation behavior of zMx α of PCHGLA beams in nonKolmogorov turbulence with α, l0 , L0 , and N is similar, but different with x00 in the region x00 ≤ 1. The difference of turbulence distance between coherent and incoherent combination cases is smaller for the PCHGLA with enough large N and x00 . According to the results obtained in the paper, the turbulence distance of GSM linear array beams, fully coherent HG array beams, and a single PCHG beam in non-Kolmogorov turbulence can be investigated easily as special cases of PCHGLA beams.
ACKNOWLEDGMENTS This work is supported by the National Natural Science Foundation of China under Grant No. 61275203, the National Natural Science Foundation of China and Civil Aviation Administration of China under Grant No. 61079023, the Science Foundation of Sichuan Provincial Education Department under Grant No. 12ZA203, the Program for Bureau of Yibin City Science and Technology under Grant No. 2013SF020, the Program for Excellent Youth Talents in Sichuan University under Grant No. 2011-2-B17, and the Program for Innovation Team of Sichuan Provincial Education Department under Grant No. 13TD0048.
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