Phase discontinuities induced scintillation enhancement: coherent vortex beams propagating through weak oceanic turbulence
Hantao Wang, Huajun Zhang, Mingyuan Ren, Jinren Yao, Yu Zhang
aa r X i v : . [ phy s i c s . op ti c s ] M a r ARTICLE TEMPLATE
Phase discontinuities induced scintillation enhancement: coherentvortex beams propagating through weak oceanic turbulence
Hantao Wang a , Huajun Zhang a , Mingyuan Ren a , Jinren Yao a and Yu Zhang a a School of Physics, Harbin Institute of Technology, No.92, Xidazhi Road, Harbin 150001,China
ARTICLE HISTORY
Compiled March 5, 2021
ABSTRACT
Under the impact of an infinitely extended edge phase dislocation, optical vortices(screw phase dislocations) induce scintillation enhancement. The scintillation indexof a beam consisting of two Gaussian vortex beams with ± KEYWORDS
Scintillation; phase dislocation; vortex beam; oceanic turbulenc.
1. Introduction
When a beam propagates through turbulence, the fluctuation of refractive index willmodify the complex amplitude of the beam and result in optical turbulence. If the tur-bulence is strong enough, some points with zero-amplitude and undetermined phasewill appear in pairs which called branch points. Tracing around a point counterclock-wise, there will be a continuous 2 π phase increase (decrease) which makes the point bea phase singularity with positive (negative) charge. Besides, each pair of branch pointswith opposite topological charges are the origins of a phase dislocation line which iscalled branch cuts. Nye and Berry initially observed this phenomenon in wave trainsand introduced the conception into optics [1]. Since then, phase discontinuities havebeen studied widely in theory and experiment, especially in the areas of atmosphereturbulence and adaptive optics. Fried and Vaughn elaborated the difference betweenthe phase discontinuity at the dark rings of the Airy diffraction pattern and branchcuts that the phase discontinuity step size of the former is π , or π +2 kπ rather than 2 π of the latter [2]. Based on these obvious contrasts, phase discontinuities can be dividedinto three main types: infinitely extended edge dislocation, screw dislocations, and lim-ited edge dislocation [3]. The first one is a classical dislocation in far-field diffraction CONTACT Yu Zhang. Email: [email protected] atterns and the laser beams with transverse cavity modes [4]. The third one can beregarded as a mixed type and it is unstable under the influence of perturbations. Bycontrast, for screw dislocations, more novel characters drive researchers to pay moreattention to its evolution behavior in free space or under different perturbations.Initially the propagation dynamic of screw dislocations, also called optical vorticesin optics [5], affected by several optical structures (e.g., optical vortices with the sameor opposite topological, background field with phase gradient or amplitude gradient)in free space has been investigated [6]. Then with the dramatic development of opticalcommunication system and Lidar in atmosphere and ocean, the propagation behaviorof optical vortices in turbulence becomes a popular topic [7]. The focus is switchedto the dynamical behavior in external perturbations. For optical communication, theultimate objective is reducing the influence of turbulence and promoting the stabilityof optical signal [8]. Several types of laser beams with optical vortices have beenshown to be less affected by turbulence when compared with non-vortex beams [9–11].However, for Lidar to measure turbulence, optical vortex becomes a new parameter tocharacterize some aspects of turbulence. Ref. [12] derived the theoretical expression ofdislocation density in various turbulent and propagation conditions. Then, numericalexperiments were carried out to prove that singularity density increases slowly with theincrease of Rytov index which represents the intensity of turbulence [13]. Therefore,the singularity density may be used to depict the intensity of turbulence [14].Looking back on these researches, those who focused on the evolution behaviorsof screw dislocations affected by initial optical fields or external perturbations accel-erate the development of various applications of optical vortices in adaptive optics[15], optical vortex field manipulation [16] and optical measurement [17]. Further re-searches on the joint influence aimed to reduce the impact of external perturbationsthrough exploring appropriate forms of initial optical fields [18]. Thus far, the inter-action of a pair of branch points through distributed turbulence has been unknown,especially when they are infinitesimally close together [19]. And the discussion aboutthe combination impact of screw dislocations and infinitely extended edge dislocationson optical fields under perturbations has not been published yet. Therefore, in thispaper, referring to the structure of branch points and the stability of optical vortices[20], a pair of coherent Gaussian vortex beams with opposite single-charged opticalvortices were considered. An infinitely extended edge dislocation can be obtained byfully overlapping these two beams. When these two beams are partial overlapping,both an infinitely extended edge dislocation and screw dislocations coexist except forthe condition that the phase difference of two beams is π . Besides, scintillation, asa phenomenon describing the intensity fluctuations of a beam propagating throughturbulence, was chosen to reflect the behavior of two types of phase discontinuities.That is not only because the evolution of phase discontinuities can induce the intensityvariation of a beam, but also because scintillation index has been widely applicated incharacterizing the intensity of turbulence.In Section 2, we derived the scintillation index of the beam through oceanic tur-bulence and, to verify the applicability of analytical derivation, discussed that in twocases: completely overlapping (only an infinitely extended edge dislocation exists) andcompletely separating (only screw dislocations exist). Next, in Section 3, we consideredthe partial overlapping condition containing both two types of phase discontinuities topresent the formation conditions and the characters of enhanced scintillation. Then inSection 4, we used phase screens method to demonstrate the phenomenon mentionedin Section 3 and verified the theoretical analysis. In the end, we discussed the resultsof Section 3 and Section 4 in Section 5. 2 . The scintillation of coherent beams with screw dislocations or aninfinitely extended edge dislocation For the simplicity of discussion, the scintillation of the optical field consisting of twocoherent Gaussian vortex beams with ± U (cid:0) ρ ′ , z = 0 (cid:1) = U + (cid:0) ρ ′ − d , z = 0 (cid:1) + U − (cid:0) ρ ′ + d , z = 0 (cid:1) exp ( iφ ) , (1)where U ± (cid:0) ρ ′ , z = 0 (cid:1) = E ± exp − ρ ′ σ ! (cid:0) ρ ′ x ± iρ ′ y (cid:1) , (2)and ρ ′ ≡ (cid:0) ρ ′ x , ρ ′ y (cid:1) is a two dimensional vector at the plane z = 0, φ is the initialphase difference of two beams and E ± is the electric field amplitude of beams with ± E . The degree of overlap is determined jointly bythe distance of the centers of two beams 2 d and the waist width of the Gaussianbackground beam envelope σ . When d is equal to zero or approaches to infinite, thestate can be regarded as fully overlap or complete separation, respectively. Except thesetwo limiting cases, an infinitely extended edge dislocation and two screw dislocationswith opposite sign coexist in different distributions. For simplification and symmetry,the direction of separation is chosen to be x-axis and the origin of coordinates isestablished at the middle point of the centers of two beams.The general expression of the scintillation index of a beam through random mediais described as follows [21]: σ I = (cid:10) I ( ρ , z ) (cid:11)(cid:10) I ( ρ , z ) (cid:11) − , (3)where I ( ρ , z ) and I ( ρ , z ) denote the instantaneous intensity and the square of theinstantaneous intensity at the plane z with transverse coordinates ρ ≡ ( ρ x , ρ y ). h·i represents the ensemble average. It is obvious that the second moment and fourth mo-ment of optical field should be derived, respectively, before calculating the scintillationindex. Therefore, we first present the second-order statistics of optical field.In order to simultaneously characterize the variance of intensity and the evolu-tion behavior of phase discontinuities, the extended Huygens-Fresnel principle andthe cross-spectral density method are used. The cross-spectral density function at thereceiving plane can be expressed as [22] W ( ρ , ρ , z ) = k E π z Z Z U ∗ (cid:0) ρ ′ , (cid:1) U (cid:0) ρ ′ , (cid:1) D exp (cid:2) ψ ∗ (cid:0) ρ ′ , ρ , z (cid:1) + ψ (cid:0) ρ ′ , ρ , z (cid:1)(cid:3)E × exp (cid:26) − ik z h(cid:0) ρ ′ − ρ (cid:1) + (cid:0) ρ ′ − ρ (cid:1) i(cid:27) d ρ ′ d ρ ′ , (4)where k = 2 π / λ is the wave number that is related to the wavelength λ , ψ is the randompart of the complex phase of a spherical wave induced by turbulence, the asterisk ∗ denotes the complex conjugate. The ensemble average part in Eq. (4) contains the3essel function of the first kind and zero order. And Ref. [23] discussed the requiredcondition of its simplification. The approximation can be expressed as D exp (cid:2) ψ ∗ (cid:0) ρ ′ , ρ , z (cid:1) + ψ (cid:0) ρ ′ , ρ , z (cid:1)(cid:3)E ≈ exp (cid:26) − k zT h(cid:0) ρ ′ − ρ ′ (cid:1) + (cid:0) ρ ′ − ρ ′ (cid:1) ( ρ − ρ ) + ( ρ − ρ ) i(cid:27) , (5)and T = π Z ∞ κ Φ n ( κ ) d κ. (6)Here, Φ n ( κ ) is the wide-range Prandtl/Schmidt number power spectrum of refractiveindex fluctuations. It can be expressed as the linear combination of temperature spec-trum Φ T ( κ ), the salinity spectrum Φ S ( κ ) and the co-spectrum Φ TS ( κ ) in the form of[24] Φ n ( κ ) = A Φ T ( κ ) + B Φ S ( κ ) + 2 AB Φ TS ( κ ) , (7) A and B are the linear coefficients related to average temperature h T i and averagesalinity concentration h S i . Each spectrum can be written as [24]Φ i ( κ ) = h . κη ) . c i . − . κη ) . c i . i × π βε − κ − χ i exp h − . κη ) c i . i . i ∈ { T , S , TS } (8)By the way, owing to the distribution of Φ n ( κ ), although the approximation does notsatisfy the requirements in Ref. [23], Eq. (5) is still valid with the restrictive conditionsthat the intensity of turbulence is extremely weak and the transverse scale of the beamis small. This required condition should also be satisfied in the subsequent parts. InEq. (8) the Kolmogorov microscale η is defined as [25] η = v ε − , (9)where v is the momentum diffusivity and ε is the energy dissipation rate. The dimen-sionless parameters c i ( i ∈ { T , S , TS } ) are [25] c T = 0 . β P r − , c S = 0 . β Sc − , c TS = 0 . β P r + Sc P rSc , (10)where
P r is the Prandtl number, Sc is the Schmidt number and β is the Obukhov-Corrsin constant that is equal to 0.72. In Eq. (8) χ i ( i ∈ { T , S , TS } ) are the ensemble-averaged variance dissipation that can be defined by [25] χ T = K T (cid:18) d h T i dz (cid:19) , χ S = K S (cid:18) d h S i dz (cid:19) = d r H χ T ,χ TS = K T + K S (cid:18) d h T i dz (cid:19) (cid:18) d h S i dz (cid:19) = 1 + d r H χ T , (11)4here K T and K S are the eddy diffusivity of temperature and salinity, respectively.The eddy diffusivity ratio d r is derived from density ratio R ρ = α | H | / β . And it canbe written as [25] d r ≈ R ρ + R . ρ ( R ρ − . , . R ρ − . , . R ρ , R ρ ≥ , . ≤ R ρ < ,R ρ < . . (12)The H represents the temperature-salinity gradient ratio, α and β are the thermalexpansion coefficient and saline contraction coefficient, respectively. The intensity ofturbulence can be determined only by ε , H , χ T , h T i and h S i . For intuitive charac-terization, we prefer using Rytov variance σ R (the scintillation for a plane wave) asa recognized indicator to describe the fluctuation conditions of oceanic turbulence.Therefore, the analytical expression of Rytov variance is derived as follows: σ R = 8 π k L Z Z ∞ κ Φ n ( κ ) (cid:20) − cos (cid:18) Lκ ξk (cid:19)(cid:21) d κ d ξ = A σ R T + B σ R S + 2 ABσ R TS , (13)with σ Ri = π k Lβ ε − χ i η X j =1 b j (cid:0) . c i . (cid:1) − a j ( a j − a j − × Γ (cid:18)
76 + a j (cid:19) + 66 a j − (cid:0) θ − i (cid:1) − aj Γ (cid:18) −
56 + a j (cid:19) × ( cos (cid:20)(cid:18) − a j (cid:19) arccot ( θ i ) (cid:21) + θ i sin (cid:20)(cid:18) − a j (cid:19) arccot ( θ i ) (cid:21))) ,i ∈ { T , S , TS } , (14)and b j = (cid:0) . c i . − . c i . (cid:1) , a j = (cid:0) .
612 0 . (cid:1) , (15)where θ i = (cid:0) . c i . (cid:1) η k (cid:14) L and Γ is the Gamma function.Return to the derivation of cross-spectral density, the analytical expression of T based on Φ i ( κ ) is obtained. Similar to the derivation of Rytov variance in Eq. (13), T is also able to transform into the linear combination of T i ( i ∈ { T , S , TS } ) that canbe written as T i = π β ε − χ i η − X j =1 b j (cid:0) . c i . (cid:1) − − a j Γ (cid:18)
16 + a j (cid:19) . (16)Note that, for simplicity, the parameters associated with Φ n ( κ ) in the rest of thispaper are just presented by the components associated with Φ i ( κ ). Then, according5o the integral formula [26] Z ∞−∞ x n exp (cid:0) − px + 2 qx (cid:1) d x = n ! exp (cid:18) q p (cid:19) r π p (cid:18) qp (cid:19) n ⌊ n /2 ⌋ X k =0 n − k )! ( k )! (cid:18) p q (cid:19) k , (17)the result of Eq. (4) is shown as follows W ( ρ , ρ , z ) = k E z p p exp (cid:18) − d σ (cid:19) exp h − k zT ( ρ − ρ ) i exp (cid:18) − ik ρ − ρ z (cid:19) × exp q y p + q y p ! ( S ++ + S −− + C + − + C − + ) , (18)with S ±± ( ρ , ρ , z ) = exp " ( q x ± D x ) p + ( q x ± D x ± D x ) p ( q x ± D x ± dp ∓ iq y ) × ( q x ± D x ± D x ± dp ± iq y ) + k zT + k zT × " p ( q x ± D x ± D x ) + q y p ± d ( q x ± D x ± D x ∓ iq y ) , (19) C ±∓ ( ρ , ρ , z ) = exp ( ± iφ ) exp " ( q x ± D x ) p + ( q x ∓ D x ± D x ) p × ( ( q x ± D x ± dp ∓ iq y ) ( q x ∓ D x ± D x ∓ dp ∓ iq y ) + k zT × (cid:20) p ( q x ∓ D x ± D x ∓ iq y ) ∓ d ( q x ∓ D x ± D x ∓ iq y ) (cid:21)) , (20)where p = 1 σ + k zT + ik z , p = 1 σ + k zT − ik z − k z T p , D x = − dσ , D x = − k zT dσ p , q = 12 k zT ( ρ − ρ ) + ik z ρ , q = − k zT ( ρ − ρ ) − ik z ρ + q k zTp , (21) q ≡ ( q x , q y ) and q ≡ ( q x , q y ). S ±± represents the cross-spectral density of singlebeams with ± C ±∓ is the cross-term of two beams. The h I i in Eq. (3) can be obtained when ρ = ρ . In addition, the evolution behavior of phasediscontinuities can be obtained based on spectral degree of coherent which is defined6s [27] µ ( ρ , ρ , z ) = W ( ρ , ρ , z ) p I ( ρ , z ) I ( ρ , z ) . (22)And the position of optical vortex is determined by [28]Re [ µ ( ρ , ρ , z )] = 0 , Im [ µ ( ρ , ρ , z )] = 0 , (23)where Re denotes the real part and Im represents the imaginary part.After completing the derivation of the second-order statistics, we set about dis-cussing the fourth-order statistics of optical field. The general fourth-order cross-coherence function at receive plane can be expressed in the form of [21] W ( ρ , ρ , ρ , ρ , z ) = k π z Z Z Z Z U (cid:0) ρ ′ , (cid:1) U ∗ (cid:0) ρ ′ , (cid:1) U (cid:0) ρ ′ , (cid:1) U ∗ (cid:0) ρ ′ , (cid:1) × D exp (cid:2) ψ (cid:0) ρ , ρ ′ , z (cid:1) + ψ ∗ (cid:0) ρ , ρ ′ , z (cid:1) + ψ (cid:0) ρ , ρ ′ , z (cid:1) + ψ ∗ (cid:0) ρ , ρ ′ , z (cid:1)(cid:3)E exp (cid:26) ik z h(cid:0) ρ ′ − ρ (cid:1) − (cid:0) ρ ′ − ρ (cid:1) + (cid:0) ρ ′ − ρ (cid:1) − (cid:0) ρ ′ − ρ (cid:1) i(cid:27) d ρ ′ d ρ ′ d ρ ′ d ρ ′ , (24)where D exp (cid:2) ψ (cid:0) ρ , ρ ′ , z (cid:1) + ψ ∗ (cid:0) ρ , ρ ′ , z (cid:1) + ψ (cid:0) ρ , ρ ′ , z (cid:1) + ψ ∗ (cid:0) ρ , ρ ′ , z (cid:1)(cid:3)E = exp h E (0 ,
0) + E (cid:0) ρ − ρ , ρ ′ − ρ ′ (cid:1) + E (cid:0) ρ − ρ , ρ ′ − ρ ′ (cid:1) + E (cid:0) ρ − ρ , ρ ′ − ρ ′ (cid:1) + E (cid:0) ρ − ρ , ρ ′ − ρ ′ (cid:1) + E (cid:0) ρ − ρ , ρ ′ − ρ ′ (cid:1) + E ∗ (cid:0) ρ − ρ , ρ ′ − ρ ′ (cid:1)i , (25)and E (0 ,
0) = − π k Z L Z ∞ κ Φ n ( κ )d κ dz , (26) E ( ρ , ρ ) = 4 π k Z L Z ∞ κ Φ n ( κ, z ) J { κ | ( ρ − ρ ) |} d κ dz , (27) E ( ρ , ρ ) = − π k L Z Z ∞ κ Φ n ( κ, z ) J { κ | ( ρ − ρ ) |}× exp (cid:26) − iLκ k (1 − ξ ) ξ (cid:27) dκdξ, (28) J ( x ) is the first kind and zero order of the Bessel function, ξ = 1 − z / L is the normal-ized distance variable. The analytical derivation of W ( ρ , ρ , ρ , ρ , z ) is complicated7nd only (cid:10) I (cid:11) is what we really want. Therefore, Eq. (24) is reasonable to be simplifiedinto the form where ρ = ρ = ρ = ρ = ρ . Combining the approximation used inEq. (5) and the simplification method, the calculations of Eqs. (26–28) are able totransform into E (0 ,
0) = − π k LT , E (cid:0) ρ , ρ , ρ ′ , ρ ′ (cid:1) ≈ π k LT − k LT (cid:0) ρ ′ − ρ ′ (cid:1) ,E (cid:0) ρ , ρ , ρ ′ , ρ ′ (cid:1) ≈ − π k LT + π k LT (cid:0) ρ ′ − ρ ′ (cid:1) , (29)and the analytical expressions of T , T and T are presented as follows T i = 18 π β ε − χ i η X j =1 b j (cid:0) . c i . (cid:1) − a j Γ (cid:18) −
56 + a j (cid:19) , (30) T i = 14 π β ε − χ i − X j =1 " b j (2 η ) a j Γ (cid:18) a j − (cid:19) (cid:18) × . η c i . + i Lk (cid:19) − a j × F (cid:18) , −
56 + a j ; 32 ; iLiL + 4 k . η c i . (cid:19)(cid:21) , (31) T i = 14 π β ε − χ i X j =1 b j η a j Γ (cid:18) a j − (cid:19) " (cid:18) × . η c i . + i Lk (cid:19) − × (cid:0) . η c i . (cid:1) − a j − ik L (cid:0) . η c i . (cid:1) − a j + 4 a j − L − × (cid:18) × . η c i . + i Lk (cid:19) − − a j (cid:18) ik . η c i . + 6 a j − L (cid:19) × F (cid:18) , −
56 + a j ; 32 ; iLiL + 4 k . η c i . (cid:19)(cid:21) , (32)where F is the hypergeometric function. The analytical expression of (cid:10) I (cid:11) is able toobtained in accordance with the procedure used in second-order statistics derivationthrough onerous calculations. However, the derivation results are too complicated tobe presented in this paper. We just substitute these results into the scintillation indexand present the numerical results of scintillation index with different situations.In order to verify the applicability of the analytical derivation result obtained, wechoose two general cases to calculate their scintillation indexes. Here, we discuss thefirst case corresponding to the condition of d → ∞ that only a screw dislocation exists.In other words, the scintillation of a Gaussian vortex beam with single charged and theevolution behavior of optical vortex are investigated. In this case, there is no additionalbackground filed to influence the propagation of the optical vortex. Therefore, we justexpound the scintillation index with different intensities of oceanic turbulence andhow an optical vortex evolves. The variables associated to oceanic turbulence are setto be fixed values except χ T to achieve the most convenient way of changing theintensity of oceanic turbulence. The fixed parameters are set as: ε = 10 − m s − , H = − ◦ C · ppt − , h T i = 15 ◦ C and h S i = 34 . χ T and the parameters of the initial beam should be determined under8he consideration of the approximation of J ( x ) from power series expansion and theapplicability of approximate method used in Rytov variance. So the transverse size ofthe initial beam is set to be about the order of magnitude of 10 − m, χ T is rangingfrom 0 to 10 − K s − and the transmission distance is chosen to be 5m. In this case,the Rytov variance is capped at 2 . × − . -10 -9 -8 S c i n till a ti on I nd e x c T (K s -1 ) s = 5 mm s = 4 mm s = 3 mm s = 2 mm s = 1 mm Figure 1.
The evolution behavior of on-axis scintillation index of Gaussian vortex beam with single chargedthrough oceanic turbulence for different σ and χ T . The on-axis scintillation indexes of the Gaussian vortex beam whose wavelengthis 532nm with single charged versus the intensity of weak oceanic turbulence areillustrated for different σ in Figure 1. The variation of scintillation indexes for severalvalues of σ are small under perturbation. And the variation range is in agreement withthat in Ref. [29]. Therefore, the approximation in Eq. (5) is valid in these conditions.The smaller the σ is, the steeper the descending portion of the scintillation index curvewill be. That is caused by the rapid raise of the on-axis ensemble average intensitythat reflects the sensitivity of the beam to oceanic turbulence. The on-axis scintillationindex of the Gaussian vortex beam is widely different from that of the Gaussian beamin weak oceanic turbulence for its large scintillation index. But for resemblance, thesebeams still retain weak response to the variation of the intensity of oceanic turbulence. -10 -9 -8 S c i n till a ti on I nd e x c T (K s -1 ) s = 5 mm s = 4 mm s = 3 mm s = 2 mm s = 1 mm Figure 2.
The evolution behavior of on-axis scintillation index of a beam that only contains an infinitelyextended edge dislocation through oceanic turbulence for different σ and χ T . The second case is the condition of d = 0 that only an infinitely extended edgedislocation exists. The optical field of this case can be regarded as one of two mutuallyorthogonal components of the light field in the first case with the same distributionat z plane. Because of the statistical homogeneity and isotropy of oceanic turbulence,the scintillation index of any component is the same. Therefore, the variation of scin-9illation index is the same as that in the first case. The evolution behavior of on-axisscintillation index for d = 0 is presented in Figure 2. The trend of the curve is inaccord with the prediction before.
3. Scintillation enhancement induced jointly by two types of phasediscontinuities
In this section, the transition state that screw dislocations and an infinitely extendededge dislocation coexist is investigated and the scintillation enhancement with detaileddescription is presented by graphs. The local maximums of scintillation indexes andthe distance of two optical vortices are illustrated for different overlap ratios in Figure3(a). To make it more intuitive, the scintillation index distributions and the phasedistributions are shown for d = 0 . d = 0 . d = 0 . σ = 1mm and χ T on the transmission path is 10 − K s − . And the Rytov variance is 2 . × − . Withthe increase of the separation distance of two Gaussian vortex beams, an enhancedpeak of scintillation index appears on the scintillation index ridge. Then the peak valueincreases sharply to the maximum where d = 0 . − . , . . , . d which is the separation distance of two Gaussianvortex beams. This trend is similar to the relation of intra-creation pair separationto propagation distance in Ref. [19]. The variation of the separation distance of twooptical vortices reflects the attraction effect of the background field that caused bythe infinitely extended edge dislocation indirectly. That is similar to the dynamicalbehavior of optical vortices in Ref. [31] and Ref. [32].To verify that the combination effect of two types of phase dislocations does inducethe scintillation enhancement, we calculate the transmission results of the beam atreceive plane with only screw dislocations existing. In a similar way, we plot the curvesof the local maximums of scintillation indexes and the distance of two optical vorticeswhen φ = π in Figure 4.The scintillation index of the single beam almost keeps invariant and has nearlythe same value of that in Figure 1. When these two optical vortices are separated,the scintillation index peak mainly manifests as the independence of a single Gaussianvortex beam. But the variation of the distance of two optical vortices reflects a strongimpact from the background field. The difference between the distance of vortices (redsolid line in Figure 4) and the separation distance of two beams (red dashed line inFigure 4) indicates that the closer the vortices are, the stronger the repulsion effect will10 .0 0.5 1.0 1.50.00.51.01.52.02.53.0 enhanced peak single beam distance of vortices d (mm) S c i n till a ti on I nd e x (a) D i s t a n ce ( mm ) y = 2d (b) (d)(c) Figure 3.
The evolution behaviors of the scintillation index of a beam with two types of phase discontinuitiesand the distance of two optical vortices the beam contains. (a) The local maximums of scintillation indexesand the distance of two screw dislocations for different values of d . The scintillation indexes and the phasedistributions for (b) d = 0 . d = 0 . d = 0 . be. In fact, the change of the relative phase of two Gaussian vortex beams results inthe disappearance of the infinitely extended edge dislocation. That makes the intensitygradient of the background field be the opposite state to the former case. And it altersthe attraction effect to the repulsion effect on two optical vortices.Another condition that only an infinitely extended edge dislocation existing hasalready been discussed in section 2. Therefore, through comparing the scintillationindexes in Figure 2, Figure 3(a) and Figure 4, the coexistence of screw dislocationsand an infinitely extended edge dislocation results in the scintillation enhancement,and neither of these two types of dislocations can induce this phenomenon withoutthe other.There are two aspects mainly influencing the formation of scintillation enhancement.For the first aspect, the evolution behavior of optical vortices is the direct cause ofscintillation enhancement. The spectral degree of coherent represents the expectationof the phase distribution which is shown in Figs. 3(b)-3(d). And it also charaterizesthe probability of the annihilation of two optical vortices with opposite topologicalcharges. For example, when d < . .0 0.5 1.0 1.50.00.51.01.52.02.53.0 single beam distance of vorticesd (mm) S c i n till a ti on I nd e x D i s t a n ce ( mm ) y = 2d Figure 4.
The evolution behaviors of the local maximum of scintillation index and the distance of two opticalvortices for only screw dislocations existing in a beam. dislocation in any statistical sample. It just represents that the annihilation is in adominant position. Therefore, a critical state must exist. And the creatiom amd theannihilation of optical vortices are evenly matched at this state. In addition, it isknown that the zero amplitude points of a field are referred to as phase singularities[33]. The creation or annihilation of optical vortices can cause obvious change in theintensity of the optical field in the neighborhood. And scintillation index changes whenthese two processes are nonnegligible under weak perturbations. Above all, the criticalstate may point to the state where scintillation index reaches the maximum, as shownin Figure 3(b) and Figure 3(c). As the state of the optical field moves away from thecritical state, the enhanced scintillation fades away gradually. It can be seen clearly inFigure 3(a). Another aspect is that the background of optical field can influence thedynamical behavior of optical vortices [6]. For example, the intensity valley inducedby an infinitely extended edge dislocation in Figure 3(a). And the inverse example isshown in Figure 4. Besides, the variation of the background of an optical field due tooceanic turbulence has further influence on the dynamical behavior of optical vortices. -10 -9 -8 T (K s -1 ) S c i n till a ti on I nd e x D i s t a n ce ( mm ) -9 -9 Maximum = 3.1616
Figure 5.
The evolution behaviors of the local maximum of scintillation index and the distance of two opticalvortices for only screw dislocations existing in a beam.
After verifying the formation condition of scintillation enhancement, the evolutionbehavior under the impact of oceanic turbulence for different intensity has been stud-ied. The parameters of oceanic turbulence are set to be the same as the former except χ T varying from 10 − K s − to 2 × − K s − . The σ is still set to be 1mm and theseparation distance of two beams is selected to be 0 . χ T . It can be seen that the curve of scintillation index increasesrapidly in the region of (cid:2) − , − (cid:3) K s − and decelerates gradually before reachingthe maximum value of 3 . χ T = 7 . × − K s − . Then, the curve begins todecrease with the appearance of optical vortices. We suppose the appearance of theindependence of optical vortices is caused by the decrease of the steep degree of theintensity valley. That is to say, the evolution behavior of background optical field fordifferent intensities of oceanic turbulence influences the behavior of optical vortices.And it is shown by the change of scintillation index. This saturation phenomenon issimilar to the phenomenon in Figure 3(a) but not the scintillation saturation in strongfluctuations.Before reaching the saturation state, the variation of scintillation index is obviouslylarger than that for a single Gaussian vortex beam. And in this circumstance, theRytov variance is in the region about the order of magnitude of (cid:2) − , − (cid:3) . In otherwords, the variation of scintillation index of the enhanced peak is about five orders ofmagnitude larger than that of a plane wave.
4. The phase screen simulation of a beam with two types of phasediscontinuities
In section 3, the scintillation enhancement is presented in theory. To further verifythis interesting phenomenon, in this section, we refer to the phase screen method inRef. [34] to demonstrate the propagation process of the beam containing two types ofphase discontinuities through weak oceanic turbulence. The parameters are set to bethe same as that in Figure 3(a) for convenient comparation. For the setup of simulation,five phase screens with the size of 8mm × ×
512 points of analysis. And the split-stepbeam propagation method is used to simulate the propagation.The evolution behavior of the scintillation index of the beam for different values ofseparation distance of two Gaussian vortex beams at receive plane is investigated. Thedistributions are shown in Figs. 6(a)-6(d). The scintillation index ridge of the region ofthe infinitely extended edge dislocation agrees with that plotting in Figure 2. And itis stable with the variation of the separation distance. The scintillation enhancementcan be seen clearly in Figs. 6(b) and 6(c). The variation trend is almost the same asthat in Figure 3(a) but with tiny difference. (a) (d)(c)(b)
Figure 6.
The phase screen simulation results of the conditions in Figure 3 for several values of the separationdistance of two Gaussian vortex beams. (a) d = 0 . d = 0 . d = 0 . d = 0 . Each case in Figure 6 is the ensemble average of 500 realizations because the fur-ther increase of realizations has less impact on the error reduction. According to thescintillation index of enhanced peak in Figure 3(a), the error is within 16 .
5. Conclusion