張 瀅，馬超群，朱時(shí)軍，劉曉旭，蔡 和，安國斐，王 浟

（1. 西南技術(shù)物理研究所，四川 成都 610041；2. 南京理工大學(xué) 理學(xué)院，江蘇 南京 210094）

1 Introduction

In the process of free-space transmission, the laser beam will be affected by atmospheric turbulence, resulting in beam drift, light intensity scintillation and other phenomena that may degrade the beam transmission quality. Therefore, suppressing the influence of atmospheric turbulence through light field regulation has become an important research subject[1-5]. Researchers have found that the suppression effect of partially coherent light on atmospheric turbulence is stronger than that of completely coherent light. The theory of partially coherent light with twisted phase has been developed rapidly in recent 20 years. The concept of twisted phase was originally proposed by Simon and Mukunda et al. in 1993, and could not be decomposed into the product of two one-dimensional coordinate parameters at that time. Meanwhile, they theoretically constructed the theoretical module of twisted Gaussian Schell-model beam, and analyzed the transmission characteristics of this new beam[6]. In 1994, Friberg et al. combined previous theories with the first experiment in this regard to obtain the twisted Gaussian Schell-model beam, but their experimental optical path was complex with poor repeatability[7]. Because the magnitude of twisted-phase factor was limited by positive definiteness, not all beams could carry the twisted phase. Since then, the research has been limited to the common coherent beams. It was not until 2007 that the situation changed. In 2007,Gori et al. proposed the positive-definiteness criteria for the theory of reproducing kernel in Hilbert space, which provided an important theoretical support for the exploration of coherent structure in new stochastic light field[8]. Later, Borghi and Gori proposed the twisting conditions of axisymmetric Schell-mode correlated beam in 2015[9]. Then in 2018, they proved that the beam twisting was only related to the coherent structure of light field. They also provided the criteria for determining whether the coherent structure could carry a twisted phase[10-11]. By constructing a new coherent structure to regulate the phase, the anti-turbulence capability of the twisted beam can be further enhanced. In addition, the beam can carry orbital angular momentum[12-13], thus possessing an important application prospect in the fields such as free space optical communication, optical imaging and nonlinear optics[14-16]. As a novel beam with special correlation structure, Laguerre-Gaussian correlated (LGC)beam has received extensive attention from researchers in the application fields such as free space optical communication and optical capture. To study the suppression effect of a laser beam with special correlation structure and twisted phase on atmospheric turbulence, the Twisted Laguerre-Gaussian Correlated (TLGC) beam was specially chosen by this study to deduce the cross spectral density function and beam quality factor of this laser beam in atmospheric turbulence and to analyze its transmission characteristics.

2 Theoretical derivation

2.1 Derivation process

From the cross spectral density function at the source point of TLGC beam as well as the expanded Huygens-Fresnel diffraction integral formula and the atmospheric turbulence model, we derived the cross spectral density function and beam quality factorM2of the propagating beam. To visually display the theoretical derivation process, the flow chart of theoretical derivation is given below, as shown in Fig. 1.

Fig. 1 Flow chart of theoretical derivation圖1 理論推導(dǎo)的流程圖

2.2 Cross spectral density function

If the beam propagates along the +zaxis and the TLGC beam is located in the source plane(z=0), the cross spectral density function of any set of position vectorsr1=(x1,y1) andr2=(x2,y2) can be expressed as:

where σ0is the beam waist width, δ0is the crosssectional coherence,Lmis standard Laguerre polynomial of them-th order,kis the wave number, μ is the distortion factor, and

Un der the paraxial approximation condition,the cross spectral density functionW(ρ1,ρ2)=〈E?(ρ1,z)E(ρ2,z)〉of a beam propagating for a distance ofzin atmospheric turbulence can be expressed as the following equation by using the expanded Huygens-Fresnel diffraction integral[17-20]:

where “＜ >” represents the ensemble average; “*”represents the complex conjugate; ρ1=(u1,v1) and ρ2=(u2,v2)constitute any set of position vectors in the receiving plane; Ψ is the random term of complex phase when the spherical wave is transmitted from the source plane to the receiving plane in atmospheric turbulence. The exponential term containing the conjugate part can be written as[21]:

where Φn(κ,α) is the power spectrum function of refractive index fluctuation of turbulent medium, κ is the two-dimensional spatial frequency, and J0is the first class of zero-order Bessel function. If κ|u(r1?r2)+(1?u)(ρ1?ρ2)|?1, J0can be approximately expressed as[22]:

According to Tosell's non-k spectrum turbulence theory, Φn(κ,α) can be expressed as[19]:

where α is the power index; κ0=2π/L0, κm=c(α)/l0;L0andl0respectively represent the outer and inner scales of turbulence;is the generalized refract ive index structure constant, in m3?α;1/(α?5),where Γ (·) is Gamma function.

The substitution of Eq. (4) and Eq. (5) into Eq. (3) can obtain:

whereHm(·) is a Hermitian polynomial of them-th order;

2.3 Beam quality factor

The beam quality factorM2is an important index to evaluate the beam transmission quality in atmospheric turbulence. Based on the second-order matrix of Wigner distribution function, the factorM2of TLGC beam is calculated below. To simplify the calculation, the coordinate substitution is carried out, namely: ρs=(ρ1+ρ2)/2 and ρd=ρ1?ρ2. Then the Eq. (2) can be written in the sum-difference form:

By using the algorithm proposed in the reference [23], we denoteandrd=. Then the Eq. (8)can be expressed as:

After substituting Eq. (6) into Eq. (2) and solving a series of integrals, the cross spectral density function of TLGC beam at the transmission distancezcan be obtained:

The Wigner distribution function of partially coherent light can be obtained through the two-dimensional Fourier transform of cross spectral density equation, i.e.:

Then1+n2+m1+m2-order matrix of the Wigner distribution function of partially coherent light is defined as[24-25]:

The factor M2of partially coherent beam after propagating in atmospheric turbulence is defined as the following second-order matrix[25]:

By substituting the equations (14), (15) and(16) into Eq. (17), the factorM2of TLGC beam after propagating in atmospheric turbulence can be obtained:

3 Numerical calculation and result

According to the theory of partially coherent light, the average light intensity of the beam arriving at the receiving surface is defined asI(ρ)=W(ρ1,ρ2). Based on the above definition, the average light intensity distribution of TLGC beam in atmospheric turbulence can be studied. When the parameters of laser beam and atmosphere are not given specifically, they can be selected as follows:

The average light intensity distribution curves of TLGC beam at different transmission distances are shown in Fig. 2. The average light intensity distribution of the beam near the source plane is Gaussian distribution. With the increase of distance, the beam is gradually hollowed due to the reciprocity between the initial coherent structure of light source and the far field intensity. When the propagation distance further increases, the average light intensity of the beam will be gradually degraded into Gaussian distribution under the influence of atmospheric turbulence. In addition, the ordermof TLGC beam will affect the rate of change of average light intensity distribution over distance. Ifm= 0,the TLGC beam will be degraded into the twisted Gaussian Schell-model beam. The higher the ordermis, the flatter the light distribution curve will be.This indicates that the higher-order beam has a stronger suppression effect on atmospheric turbulence.

Fig. 2 Average light intensity distribution of a TLGC beam in atmospheric turbulence at different propagation distances圖2 扭曲拉蓋爾—高斯光束在大氣湍流中不同傳輸距離下的平均光強(qiáng)分布

To analyze the influence of beam parameters and atmospheric parameters on turbulence suppression, we analyze the influence of the following parameters on average light intensity distribution atz=5km (see the Fig. 3): (a) power index α; (b) turbulent outer scaleL0and turbulent inner scalel0;(c) distortion factor μ; (d) cross-sectional coherence degree δ0. The influence of atmospheric parameters on beam propagation is shown in Fig. 3(a)?(b).When the power index α and the turbulent inner scalel0are smaller or the turbulent outer scaleL0is larger, the light intensity distribution will be converted into hollow distribution more quickly. In other?words, the stronger the turbulence is, the more obvious the variation trend of light intensity distribution will be. This is consistent with the conclusion in ref.[19], and also verifies the validity of the numerical calculation in this paper. The suppression effect of beam parameters on atmospheric turbulence is shown in Fig. 3(c)?(d). When the distortion factorμis increased or the cross-sectional coherence degree δ0is decreased, the transformation of light intensity distribution into hollow distribution will be slowed down, indicating that the influence of atmospheric turbulent environment on light intensity distribution will be suppressed.

Fig. 3 Average light intensity distribution of a TLGC beam in atmospheric turbulence at z =5km changing with different parameters: (a) power index α; (b) turbulent outer scale L0 and turbulent inner scale l0 ; (c) distortion factor μ and (d)cross-section coherence δ0圖3 扭曲拉蓋爾—高斯光束在z=5km 處不同參數(shù)影響下的平均光強(qiáng)分布。（a）不同功率指數(shù)α ；（b）湍流外尺度L 0和湍流內(nèi)尺度l0 ；（c）扭曲因子μ ；（d）橫截面相干度δ0

The Fig. 4 shows how the normalized factorM2of TLGC beam varies with the beam parameters and atmospheric parameters. The smaller the factorM2is, the higher the beam quality will be. From Fig. 4(a), Fig. 4(c) and Fig. 4(d), it can be found that for the beam with a higher orderm, itsM2factor changes more slowly and is less affected by turbulence. As shown in Fig. 4 (a), with the increase of the power index α, the normalized factorM2will increase rapidly at first and then decrease slowly,achieving the maximum value near α=3.11. This is because the atmospheric turbulence is strongest near α=3.11, producing the greatest effect on the beam.It can be seen from Fig. 4 (b) that the beam quality will decline with the increase of propagation distance in atmospheric turbulence. However, the increase ofM2factor can be slowed down by increasing the turbulent inner scalel0or decreasing the turbulent outer scaleL0, that is, by weakening the influence brought by atmospheric turbulence. In addition, compared with the decrease of the turbulent outer scaleL0, the increase of the turbulent inner scalel0has a more significant suppression effect on atmospheric turbulence. It is not difficult to find from Fig. 4(c) and Fig. 4(d) that with the increase of the distortion factor and the decrease of the coherence, the factorM2will be reduced and the beam transmission quality will be degraded, which is consistent with the conclusion of the previous analysis.

Fig. 4 Normalized M2-factor of a TLGC beam in atmospheric turbulence changing with different parameters: (a) power index α; (b) transmission distance z; (c) distortion factor μ; and (d) cross-section coherence δ0圖4 扭曲拉蓋爾—高斯關(guān)聯(lián)光束在大氣湍流傳輸中的歸一化M2因子隨不同參數(shù)的變化情況。（a）功率指數(shù)α ；（b）傳輸距離z；（c）扭曲因子μ ；（d）橫截面相干度δ0

4 Conclusion

Based on the expanded Huygens-Fresnel principle and the Wigner function distribution, the negative effect of TLGC beam on atmospheric turbulence is investigated and suppressed. It is found by numerical simulation that the negative effect of turbulence on beam transmission quality can be effectively reduced by appropriately increasing the distortion factor of laser beam or decreasing the coherence of initial light field. For the TLGC beam, its transmission quality is affected by both beam parameters and atmospheric turbulence parameters. The TLGC beam with higher order, larger distortion factor and lower coherence has better transmission quality in atmospheric turbulence. However, the beam transmission quality will be degraded in the atmosphere with a larger turbulent outer scale and a smaller turbulent inner scale. The conclusion of this paper has certain theoretical guiding significance for the atmospheric communication and long-distance transmission of laser beam.

——中文對(duì)照版——

1 引 言

激光在自由空間的傳輸過程中受大氣湍流的影響會(huì)出現(xiàn)光束漂移、光強(qiáng)閃爍等現(xiàn)象，這將導(dǎo)致其光束傳輸質(zhì)量下降。通過光場(chǎng)調(diào)控來抑制大氣湍流的影響已成為重要的研究課題[1-5]。研究者們發(fā)現(xiàn)，部分相干光對(duì)大氣湍流的抑制效應(yīng)比完全相干光要強(qiáng)，其中，攜帶扭曲相位的部分相干光理論在近20多年得到了快速發(fā)展。扭曲相位這一概念最初是由Simon和Mukunda等人于1993年提出的，它無法分解為兩個(gè)一維坐標(biāo)參量的乘積，他們同時(shí)理論構(gòu)建了高斯-謝爾模光束被扭曲后的理論模塊，并分析了該新型光束的傳輸特性[6]。1994年，F(xiàn)riberg等人結(jié)合前人的理論首次實(shí)驗(yàn)產(chǎn)生了扭曲高斯-謝爾模光束，但是，他們的實(shí)驗(yàn)光路復(fù)雜而且重復(fù)性較差[7]。由于扭曲相位因子的大小受正定性的限制，并非所有光束都能攜帶扭曲相位，所以，從那以后的研究一直局限于常見的相干光束。直到2007年，情況才有所改變，Gori等人提出了在Hilbert空間再生核理論的正定性判斷條件，為探索新型隨機(jī)光場(chǎng)的相干結(jié)構(gòu)提供了重要的理論支撐[8]。之后，Borghi和Gori在2015年提出了軸對(duì)稱的謝爾模關(guān)聯(lián)光束被扭曲的條件[9]，隨后他們又在2018年證明了光束能否被扭曲只與光場(chǎng)的相干結(jié)構(gòu)有關(guān)，并給出了判斷相干結(jié)構(gòu)是否能攜帶扭曲相位的判據(jù)[10-11]。通過構(gòu)造新型的相干結(jié)構(gòu)對(duì)相位進(jìn)行調(diào)控，可以使扭曲光束抗湍流的能力得到進(jìn)一步的增強(qiáng)，此外，還能使光束攜帶軌道角動(dòng)量[12-13]，這將在自由空間光通信、光學(xué)成像、非線性光學(xué)等領(lǐng)域有著重要的應(yīng)用前景[14-16]。拉蓋爾—高斯關(guān)聯(lián)（LGC）光束作為具有特殊關(guān)聯(lián)結(jié)構(gòu)的新穎光束，在自由空間光通信和光學(xué)捕獲等應(yīng)用領(lǐng)域受到了學(xué)者們的廣泛關(guān)注。本文為了研究攜帶扭曲相位的特殊關(guān)聯(lián)結(jié)構(gòu)激光光束對(duì)大氣湍流的抑制效果，特地選擇了扭曲拉蓋爾—高斯關(guān)聯(lián)（TLGC）光束，推導(dǎo)出了該激光光束在大氣湍流中傳輸?shù)慕徊婀庾V密度函數(shù)和光束質(zhì)量因子，并對(duì)其傳輸特性進(jìn)行了相關(guān)分析。

2 理論推導(dǎo)

2.1 推導(dǎo)流程

文章從扭曲拉蓋爾—高斯光束源點(diǎn)處的交叉譜密度函數(shù)出發(fā)，結(jié)合拓展的Huygens-Fresnel衍射積分公式和大氣湍流模型，推導(dǎo)得到光束傳輸過程中的交叉譜密度函數(shù)和光束質(zhì)量因子M2。為了直觀顯示理論推導(dǎo)過程，理論推導(dǎo)的流程圖如圖1所示。

2.2 交叉譜密度函數(shù)

假設(shè)光束沿z軸 的正方向傳播，扭曲拉蓋爾—高斯關(guān)聯(lián)光束在源平面z=0內(nèi)，任意一組位置矢量r1=(x1,y1)和r2=(x2,y2)所對(duì)應(yīng)的交叉光譜密度函數(shù)可表示為：

其中， σ0為光束束腰寬， δ0為橫截面相干度，Lm是階數(shù)為m的標(biāo)準(zhǔn)拉蓋爾多項(xiàng)式，k為波數(shù)，μ為扭曲因子，

利用拓展的Huygens-Fresnel衍射積分計(jì)算，在傍軸近似條件下，當(dāng)光束在大氣湍流傳輸距離為z時(shí)，其交叉譜密度函數(shù)W(ρ1,ρ2)=〈E?(ρ1,z)E(ρ2,z)〉可表示為[17-20]：

其中，“＜ >”表示系綜平均，“*”表示復(fù)共軛，ρ1=(u1,v1) 和ρ2=(u2,v2)為接收平面內(nèi)任意一組位置矢量。 Ψ為球面波在大氣湍流中從源平面?zhèn)鬏數(shù)浇邮彰鏁r(shí)復(fù)相位的隨機(jī)項(xiàng)，含有共軛部分的指數(shù)項(xiàng)可寫為[21]：

其中， Φn(κ,α)是湍流介質(zhì)折射率起伏的功率譜函數(shù)，κ為二維空間頻率，J0是第一類0階貝塞爾函數(shù)，且當(dāng) κ|u(r1?r2)+(1?u)(ρ1?ρ2)|?1時(shí)，可近似表示為[22]：

根據(jù)Tosell提出的非k譜湍流理論，Φn(κ,α)可表示為[19]：

式中， α為功率指數(shù)， κ0=2π/L0， κm=c(α)/l0，L0和l0分別代表湍流外尺度和湍流內(nèi)尺度，為廣義折射率結(jié)構(gòu)常數(shù)，單位是c(α)=[2πΓ(5?α/2)A(α)/3]1/(α?5)，這 里Γ(·)表示Gamma函數(shù)。

將式(4)和式(5)代入式(3)可得到：

將式(6)代入式(2)，做一系列積分之后，可得到扭曲拉蓋爾—高斯關(guān)聯(lián)光束在傳輸距離為z處的交叉譜密度函數(shù)：

式中，Hm(·)是 m階數(shù)的厄米 多 項(xiàng) 式，

2.3 光束質(zhì)量因子

光束質(zhì)量M2因子是評(píng)價(jià)光束在大氣湍流中的傳輸質(zhì)量的一個(gè)重要指標(biāo)，下面從Wigner分布函數(shù)的二階矩陣出發(fā)，計(jì)算扭曲拉蓋爾—高斯關(guān)聯(lián)光束的M2因子。為了簡(jiǎn)便計(jì)算，采取坐標(biāo)代換： ρs=(ρ1+ρ2)/2和 ρd=ρ1?ρ2，則式(2)可寫成和差化的形式：

結(jié)合文獻(xiàn)[23]提出的運(yùn)算方法，令rs=r′s=和，則上式可以表示為：

部分相干光的Wigner分布函數(shù)可通過對(duì)交叉譜密度方程進(jìn)行二維傅立葉變換得到，即：

部分相干光Wigner分布函數(shù)的n1+n2+m1+m2階矩陣定義為[24-25]：

部分相干光束在大氣湍流傳播后的M2因子用二階矩定義為[25]：

將式(14)、式(15)、式(16)代入到式(17)，得到扭曲拉蓋爾—高斯關(guān)聯(lián)光束在大氣湍流傳播后的M2因子為：

3 數(shù)值計(jì)算與結(jié)果分析10?14m3?a，l0=1mm，L0=1m 。

根據(jù)部分相干光理論，光束到達(dá)接收面處的平均光強(qiáng)定義為I(ρ)=W(ρ1,ρ2)。結(jié)合上述定義，可研究扭曲拉蓋爾—高斯關(guān)聯(lián)光束在大氣湍流傳輸中的平均光強(qiáng)分布特性。當(dāng)光束和大氣的參數(shù)沒有特定給出時(shí)，選取如下：λ =632.8nm，σ0=2cm，μ=0.5km?1， δ0=5mm ，m=2，α=3.3，=2.5×

扭曲拉蓋爾—高斯關(guān)聯(lián)光束在不同傳輸距離下的平均光強(qiáng)分布曲線如圖2所示。該光束在接近源平面處的平均光強(qiáng)呈高斯分布，隨著距離的逐漸增加光束出現(xiàn)了空心的現(xiàn)象，這是由于光源的初始相干結(jié)構(gòu)與遠(yuǎn)場(chǎng)強(qiáng)度之間存在著互易關(guān)系。當(dāng)傳輸距離進(jìn)一步增加時(shí)，光束的平均光強(qiáng)受到大氣湍流的影響將會(huì)逐步退化為高斯分布。此外，扭曲拉蓋爾—高斯光束的階數(shù)m會(huì)影響平均光強(qiáng)分布隨距離變化的速度。當(dāng)m= 0時(shí)，扭曲拉蓋爾—高斯關(guān)聯(lián)光束將會(huì)退化為扭曲高斯謝爾模光束。階數(shù)m越高，光強(qiáng)分布趨勢(shì)越平緩，說明高階光束對(duì)大氣湍流的抑制作用更強(qiáng)。

為了分析光束和大氣參數(shù)對(duì)湍流抑制作用的影響，圖3給出了光束在z=5km處平均光強(qiáng)分布受功率指數(shù)α、湍流外尺度L0和湍流內(nèi)尺度l0、扭曲因子μ 及橫截面相干度 δ0的影響情況。圖3（a）、3（b）顯示了大氣參數(shù)對(duì)光束傳輸?shù)挠绊?，?dāng)功率指數(shù) α和湍流內(nèi)尺度l0越小，或湍流外尺度L0越大時(shí)，光強(qiáng)分布轉(zhuǎn)換為空心分布越快。這說明湍流越強(qiáng)，光強(qiáng)分布變化趨勢(shì)越明顯，這與文獻(xiàn)[19]的結(jié)論相一致，同時(shí)驗(yàn)證了本文數(shù)值計(jì)算的有效性。圖3（c）、3（d）顯示了光束參數(shù)對(duì)大氣湍流的抑制作用。當(dāng)增大扭曲因子μ，或降低橫截面相干度 δ0時(shí)，會(huì)減慢光強(qiáng)分布轉(zhuǎn)化為空心分布的過程，說明大氣湍流環(huán)境對(duì)光強(qiáng)分布的影響將會(huì)受到抑制。

圖4給出了扭曲拉蓋爾—高斯關(guān)聯(lián)光束的歸一化M2因子隨光束和大氣參數(shù)變化的情況，M2因子越小說明光束質(zhì)量越好。從圖4（a）、4（c）、4（d）可以發(fā)現(xiàn)，階數(shù)m越高的光束，其M2因子的變化趨勢(shì)更緩慢，受到湍流的影響也更小。如圖4（a）所示，歸一化M2因子隨功率指數(shù)α 呈先急速增加后緩慢減小的趨勢(shì)，在 α=3.11附近取得最大值。這是因?yàn)樵?α=3.11附近處大氣湍流強(qiáng)度最大，光束所受的影響也最大。從圖4（b）中可以看出，在大氣湍流中，隨著傳輸距離的增加光束質(zhì)量將下降，但增大湍流內(nèi)尺度l0或減小湍流外尺度L0都能使M2因子的增加變得緩慢，即減弱大氣湍流帶來的影響。此外，增加湍流內(nèi)尺度l0相較于降低湍流外尺度L0，對(duì)抑制大氣湍流的作用更為明顯。從圖4（c）、4（d）中不難發(fā)現(xiàn)，扭曲因子的增加和相干度的降低都可以減小M2因子，光束傳輸質(zhì)量的劣化也隨之降低，這與前文的分析相一致。

4 結(jié) 論

基于拓展的惠更斯-菲涅爾原理和Wigner函數(shù)分布，研究了扭曲拉蓋爾—高斯關(guān)聯(lián)光束對(duì)大氣湍流所帶來的負(fù)面影響的抑制。通過數(shù)值模擬發(fā)現(xiàn)，適當(dāng)增加激光光束的扭曲因子或降低初始光場(chǎng)的相干度都可以有效減小湍流對(duì)激光光束傳輸質(zhì)量的負(fù)面影響。對(duì)于扭曲拉蓋爾—高斯關(guān)聯(lián)光束來說，光束傳輸質(zhì)量同時(shí)受到光束參數(shù)和大氣湍流參數(shù)的共同作用。高階數(shù)、相對(duì)較大的扭曲因子、低相干度的扭曲拉蓋爾—高斯關(guān)聯(lián)光束在大氣湍流中的傳輸質(zhì)量更好，而具有較大湍流外尺度和較小湍流內(nèi)尺度的大氣會(huì)降低光束的傳輸質(zhì)量。本論文的研究結(jié)論對(duì)激光的大氣通信和遠(yuǎn)距離傳輸具有一定的理論指導(dǎo)意義。