Physical meaning of the deviation scale under arbitrary turbulence strengths of optical orbital angular momentum
Zhiwei Tao, Yichong Ren, Azezigul Abdukirim, Shiwei Liu, Ruizhong Rao
RResearch Article Journal of the Optical Society of America A 1
Physical meaning of the deviation scale under arbitraryturbulent strengths of optical orbital angularmomentum Z HIWEI T AO , Y ICHONG R EN , A ZEZIGUL A BDUKIRIM , S HIWEI L IU , AND R UIZHONG R AO School of Environmental Science and Optoelectronic Technology, University of Science and Technology of China, Hefei 230022, China Key Laboratory of Atmospheric Optics, Anhui Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Hefei 230031, China * Corresponding author: [email protected] February 4, 2021
Studying the meaning of the deviation scale proposed by C. M. Mabena [C. M. Mabena et al ., Phys. Rev.A 99, 013828 (2019)] may facilitate us to make better decisions on when we can employ the phase pertur-bation approximation to simulate the effect of atmospheric turbulence on the orbital angular momentum(OAM) carrying beam, which offers a valuable guideline to the optical propagation procedures underspecific scintillation conditions. Thanks to the multiple phase screen (MPS) approach, we explain thephysical meaning of the deviation scale in this paper, it is demonstrated that when the turbulent channelat weak scintillation scale, slight intensity modulation can cause the vortex split into multiple individualvortices as well as generate the vortex-antivortex pairs, after a spatial accumulation over a distance, thedeviation starts to emerge when the vortex splitting ratio passes this threshold, which means that theoccurrence of such circumstance happens to be unpredictable by the scintillation index and needs to bemeasured with the help of the vortex splitting ratio. © 2021 Optical Society of America http://dx.doi.org/10.1364/JOSAA.XX.XXXXXX
1. INTRODUCTION
Photon carrying orbital angular momentum (OAM) was firstlyproven by Allen in 1992[1], which offer an alternative way toencode information in a large alphabet. It had led to impressivedemonstrations using such advantage of high dimensionalityin many applications such as communication system[2], opticaltrapping[3, 4], imaging[5, 6], remote sensing[7], and quantuminformation[8–10]. Unfortunately, since the OAM-carrying beamhas a helical wavefront associated with azimuthal phase variesfrom 0 to 2 π l , where l represents the topological charge, it be-comes fragile in free-space propagation.Recently, most previous studies, including theories[11–15]and experiments[16–19], have been devoted to the investiga-tion of turbulent effects on the OAM modes using phase distor-tion approach, i.e., single phase screen (SPS) approach. Thesemodels indicated that the measured OAM expectation valueat the receiver the same as that of the transmitted mode. Until2017 Lavery[20, 21] performed the experiment transmitting high-dimensional structured optical light in urban environments, theyfound these conclusions did not hold for long free-space linklengths or more turbulent links. Moreover, in the past year, C.M. Mabena, et al. [22]derived an analytical expression that givesdecisive prediction to describe atmospheric crosstalk in all scin-tillation conditions using the infinitesimal propagation equation (IPE) of Roux[23, 24].However, although the evolution of OAM propagatingthrough the turbulent channel has achieved a vast amount ofattention over past decades, very few efforts have been specifi-cally conducted on the meaning of the critical point associatedwith the SPS breaks down and starts to deviate from the IPE pre-diction. In another work, we called this the deviation scale[22].In the present contribution, we first examine the minimum pa-rameter dependence of the crosstalk of OAM-carrying beampropagating through the turbulent link with the help of multiple-phase screen (MPS) approach compared with the IPE theoreticalprediction, radial numbers scrambling arising from the turbulentenvironment is also considered during the calculation. Secondly,we elucidate the physical meaning of the deviation scale underweak-to-strong turbulent strength of the crosstalk probabilityevolution. The investigation of the deviation scale may be ad-vantageous for selecting a suitable approximation to simulatethe propagation problem of the OAM-carrying beam and deter-mine the scintillation condition of the turbulent channel throughthe phase perturbation.The pure phase perturbation is typically used by the assump-tion that the scintillation is weak enough the diffraction in beampropagation can be neglected; moreover, it is valid only withinthe range of geometrical optics to be established. Nevertheless, a r X i v : . [ phy s i c s . op ti c s ] F e b esearch Article Journal of the Optical Society of America A 2 It is rather ironic that the interpretation of the deviation scaleis clearly elaborated when the assumption is removed and in-tensity fluctuations are considered, which is done by using analternative solution valid in all scintillation conditions called thesplit-step beam propagation method[25]. Interestingly, it canmanifest that only a part of intensity fluctuations had led to theonset of the deviation through calculating the vortex-splittingratio.Our conclusion is achieved from the difference between thenumerical value and the SPS counterparts. When the Rytovvariance less than unity and propagation distance is below acertain threshold, the results obtained by the simulations havethe same trend with SPS values. Conversely, the deviation startsto emerge when the distance passes this threshold. On the otherhand, the beginning of the propagation was also accompaniedby the emergence of intensity fluctuation. Hence, for opticalfields propagated over a certain range under specifical turbu-lent strength, the spatial cumulative effect of turbulence willsplit the vortex into different individual vortices and generatevortex-antivortex pairs, resulting in the deviation scale that isconsistently less than or equal to the weak scintillation scale.Such phenomenon that attached with the intrinsic property:intensity fluctuation also indicated the phase perturbation ap-proximation is not suitable for OAM-carrying simulating underspecifical weak scintillation circumstance.The paper is structured as follows. We describe the theoret-ical model of the crosstalk evolution of OAM-carrying beamsin Sec. 2. In Sec. 3, we first briefly introduce different physicalquantities defined in Ref. [26], and then apply the technicalaspects of our numerical simulation as well as reexamine theminimum parameter dependence considering radial numberscrambling using the MPS method. Based on the above sectionresults, Sec. 4 presents the difference curves as a function ofscintillation index under different normalized propagation dis-tances and elaborately explains the main results of this work:the physical interpretation of the deviation of IPE predictionsfrom SPS approximation. Sec. 5 we conclude.
2. DESCRIPTION OF THE CROSSTALK EVOLUTION
Our numerical simulation system is shown in Fig. 1. Withoutloss of generality, we assume that the distance between transmit-ter and receiver is z , and the source produced an OAM-carryingbeam is a single LG mode, which can be expressed in normalizedcylindrical coordinates by LG ( q ) p , l ( r , φ , 0 ) = A (cid:18) i √ kz R rq (cid:19) | l | + L | l | p (cid:32) kz R r | q | (cid:33)(cid:18) − q ∗ q (cid:19) p e − ikr q r e il φ (1) with L | l | p ( · ) as the generalized Laguerre polynomial and p asradial numbers, where A = (cid:112) n !/ π ( | l | + n ) !is the normaliza-tion constant and q = iz R represents the complex parameterassociated with beam waist.An OAM-carrying beam propagating through the free-spacechannel in the presence of atmospheric turbulence is often de-scribed by the split-step beam propagation method[25]. It com-monly refers that free-space link can be divided into a seriesof turbulent cells. Each set of cells build up irradiance fluctua-tion by Fresnel diffraction and introduce a random contribution Fig. 1.
The source generates a single LG mode and sendsthrough atmospheric turbulence (modeled by a series oftrapezoidal phase screens) toward a detector φ to the beam phase, as shown in Fig. 1. To agree well withthe phase structure function of Kolmogorov turbulence, randomphase screen lost low spatial frequencies need to be compensatedusing subharmonic method[27].With this in mind, the pure phase perturbation approxima-tion is substituted for the wave field of propagated LG mode inthe presence of turbulence. According to the fact that only LGmodes with the same ω satisfy the orthogonality relation, themode that beamwidth with q ( z ) = z + q , where the notation q ( z ) called the Siegman complex parameter[28], is chosen as anorthogonal basis to perform mode decomposition, which meansthat the intermodal crosstalk is entirely attributed to turbulenteffects. The receiving beam after propagation in the turbulentchannel can be expressed by the superposition of different OAMmodes as U p , l ( r , φ , z ) = ∞ ∑ p = ∞ ∑ l = − ∞ c p , l LG ( q ) p , l ( r , φ , z ) with c p , l = (cid:90) ∞ rdr (cid:90) π d φ U p , l ( r , φ , z ) LG ( q ) ∗ p , l ( r , φ , z ) (2) where the asterisk denotes the complex conjugate. c p , l meansthat the received OAM spectrum arising from turbulence pertur-bation scrambled into other radial numbers and OAM modes,surrounding the initially transmitted mode. The probability offinding one photon in the sent OAM state with index l at thereceiving plane is written as P l = ∞ ∑ p = (cid:12)(cid:12)(cid:12) c p , l (cid:12)(cid:12)(cid:12) (3) From the above analysis, a final illustration of the effects ofturbulence on the OAM modes is reflected in two aspects. Onthe one hand, incident light with OAM index l convert intothe transverse modes of LG ( q ) p , l ( z ) according to the probability (cid:12)(cid:12)(cid:12) c p , l (cid:12)(cid:12)(cid:12) ; on another hand, random phase arg (cid:16) c p , l (cid:17) is attached tothe transverse modes of LG ( q ) p , l ( z ) .
3. NUMERICAL DETAILS AND RESULTS
A. Several parameters
In order to extend the conclusion of minimum parameter de-pendence considering radial numbers scrambling with MPSsimulation, it is convenient to introduce several dimensionlessquantities which are theoretically proved indispensable in the esearch Article Journal of the Optical Society of America A 3
Table 1.
Different system constants K combined withdimension parameters shown in Fig. 2 K C n (cid:16) m − (cid:17) ω ( m ) λ ( nm ) − − − × − × − × − − − − × − t = z / z R , thenormalized turbulent strength K = C n ω π / λ and the rel-ative beam width W = ω / r , where C n is so-called refractiveindex structure constant, z R = πω / λ denotes the Rayleighrange, ω , λ and r respectively stand for the wavelength, theGaussian beam width and the Fried parameter, z is propaga-tion distance. By fixing W and t , those quantities are related by W = ( Kt ) .The scintillation index is a measurement for strength ofintensity fluctuations which arise from phase distortion com-bined with Fresnel diffraction, quantified by the Rytov variance σ R = C n k z , where k = π / λ is the wave number,is import for the implementation of our numerical simulation,which can also express in terms of the dimensionless parametersas[29] σ R = W t = W / K (4) B. Data details
Numerical simulation in one iteration, providing the randomresults of each propagation, can be repeated several times. Theensemble average of different iterations with the two steps repre-sents the OAM crosstalk evolution under one system condition.Table 1 gives the different normalized turbulent strengths K used in our simulation, which is a compound quantity that in-cludes the information of turbulence and OAM-carrying beam.For the convenience of comparison, these dimensional variablesare specifically assigned to guarantee the relative beam width W is limited to a certain range. Therefore, sampling intervalsand transmission distances vary with the compound quantitiesduring this procedure.Eq. (4) gives the relationship between the scintillation in-dex and the normalized turbulent strength. By adjusting thecompound quantities with fixed relative beam width, differentscintillation conditions can be acquired during the numericalexperiments. Additionally, since the spacing between two turbu-lent cells varies smartly with these parameters, different dimen-sions of LG bases for mode decomposition are used in differentsimulations. Fig. 2.
The probability that the received OAM power remains inthe initially mode against W with different normalizedturbulent strengths K given in Table I for (a) l = l = l = l =
3, averaged over 300 realizations of turbulence.The solid lines represent the theoretical predictions; thedashed-dotted lines represent the Rytov variance equal to unity.The Rytov variance in K = l , deviations occur at asmaller value of the Rytov variance in K = C. Evolution rules under arbitrary scintillation conditions
The probability of received OAM power that remains in theinitial mode is presented in Figure 2 as a function of W for differ-ent OAM values. The solid curve in all four plots correspondsto the theory in Paterson C using quadratic approximation[30],while the results from the MPS approach are shown as differentshaped lines, corresponding to different values of K given inTable I. According to Eq. (4), four curves of the Rytov variancein K = W are provided in different az-imuthal indices, the crossing point of these numerical outcomes( K = K (cid:38)
20 for l =
0, 1 and K (cid:38)
100 for l =
2, 3, the evolution trend of OAMcrosstalk probability given by numerical simulation almost coin-cides with the exact solution provided by the Paterson C theory,but this kind of coincidence will not exist when K is small ( K (cid:46) l =
0, 1 and K (cid:46)
20 for l =
2, 3). It means that another dimen-sionless parameter K is needed to describe the evolution of OAMcrosstalk probability except for W . Moreover, the power fractionevolution with higher-order ( l (cid:38)
2) decreases faster, more notice-
Table 2.
Different combinations of dimensionless parameter K = shown in Fig. 3. set C n (cid:16) m − (cid:17) ω ( m ) λ ( nm ) − − × − × − − esearch Article Journal of the Optical Society of America A 4 Fig. 3.
The probability as a function of W with the samenormalized turbulent strength K = l =
0, averaged over 300 realizations of turbulence. Each set ofdimensionless parameters are combined with different beamand turbulent parameters. The error bars represent the standraderror.able when K (cid:46)
2, possibly since beams with higher azimuthalindices have larger second-moment radius (rms radius) that theyare more susceptible to atmospheric turbulence.The deviation of numerical simulation from the theory ofPaterson C under arbitrary turbulent strengths of the crosstalkprobability evolution is also depicted in Fig. 2. For the com-parison with Rytov variance, it is found that the crossing pointindicated the onset of strong scintillation does not coincide withthe deviation point when K = D. Minimum parameter set verification
Figure 3 shows the evolution curve of OAM crosstalk probabil-ity as a function of K under different parameter combinationswith the same normalized turbulent strength K = K is chosen because the evolution ofcrosstalk probability at this time has deviated significantly fromthe result under the condition of weak scintillation. As can be Table 3.
Different system constants t combined withdimension parameters shown in Fig. 4. t z ( km ) ω ( m ) λ ( nm ) × − × − − × − − K , which means all kinds of beam parametersand turbulence parameters always appear in the form of dimen-sionless compound quantity in the regime of the Kolmogorovtheory of weak to strong scintillation. Compared with the crit-ical parameters deciding the IPE of a single photon travelingthrough the turbulent air, the verification process makes oursimulation valid.
4. PHYSICAL MEANING OF THE DEVIATION SCALE
A. Qualitative understanding
Fig. 2 explicitly illustrates the difference between the deviationscale and the onset of the strong scintillation, but what is thephysical nature of the deviation point remains to be explored.From the above analysis, compound quantities composed with K and W can completely determine the evolution of intermodalcrosstalk induced by turbulence. Consequently, it is not neces-sary to investigate the evolution rule under different normalizedpropagation distance t . However, although these two param-eters and W are pairwise independent variables, it may helpus give a qualitative understanding of the deviation scale bystudying the influence of t .In order to better determine the scintillation index and thetransmission distance where the numerical simulation resultsstart to deviate from the theoretical limit curve, we plot the curveof the difference of OAM crosstalk probability, ∆ P = P th − P num ,between the numerical simulation and the theoretical calculationwith the increasing σ R under different normalized propagationdistances t defined in Table 3 in Fig. 4.To facilitate analysis, Fig. 4 is divided into four areas. We cansee that the points where the numerical curves start to deviatefrom theoretical counterparts occur at σ R (cid:39) Fig. 4.
The difference between numerical value and theoreticalprediction as a function of the Rytov variance with differentnormalized propagation distances t given in Table III for (a) l = l = l = l =
3, averaged over 300 realizationsof turbulence. Each graph is divided by four areas with σ R < < σ R <
1, 1 < σ R <
10 and σ R >
10. Thedifference of probability in K = t in each graph. esearch Article Journal of the Optical Society of America A 5 Fig. 5.
The probability of the incident mode keeping its own form invariant (a)-(c), the vortex-splitting ratio and the average OAM(d)-(f) as a function of W with the normalized turbulence strength K = l = l =
2; (c), (f) l =
3. The standard error is plotted by the shaded area around each numerical outcome; the solid lines representthe theoretical predictions; the Rytov variance in K = σ R < σ R > ν = l = ν = l = ν = l = < σ R < t <
1, the deviation of simulation from theoryprediction remains still very small. On the contrary, if one oftenable condition that short-range ( t < ) and weak scintilla-tion conditions are not satisfied anymore, the deviation rapidlygrow with the increase of σ R , even if the turbulent channel isstill in the weak scintillation scale at this time. To convenientlymake a qualitative interpretation, the Cruciform scatter pointrepresenting K = W can be instead of t , as a comparison with intermodal crosstalkevolution rules, it is more clearly can be seen from Fig. 4 thatwhen K = t even though the scintillation index remains to be lessthan unity, which intuitively indicates that the condition thatkeeping the SPS model established is under weak scintillationscale with short-range.Another conclusion can be achieved from four areas pre-sented in Fig. 4 without relationship to the deviation scale. Whenthe turbulent link is beyond the weak range (1 < σ R < σ R and the highest speed of the diminution always appears inthe case of the largest l . Moreover, the degree of deviation inte-grally inclines to be the same and levels off to zero when σ R > t . The mainreason likely due to the effect of saturated transmission scintilla-tion. B. the accumulation of vortex-splitting and vortex-antivortexpairs
What can be seen from the above analysis is the deviation fromthe results given by the phase perturbation model in the weakscintillation case due to the cumulative effect of the transmissiondistance at a specific turbulence intensity. One possible reasonis that this turbulence-induced light transmission effect is notmeasurable by the scintillation index; for example, the beamwandering phenomenon caused by the repeated modulation of the phase screen and Fresnel diffraction can be ignored underthe weak scintillation[31]. Nevertheless, for the vortex beam thathas undergone turbulent perturbations, the compound effecton the beam caused by multiple iterations is much more severethan the phase distortion alone. Even at the weak scintillationscale, the intensity fluctuations caused by the above process cancause the splitting of multiple vortices of the original beam intoa single vortex with OAM equal to unity and the generation ofvortex-antivortex pairs with OAM equal to + −
1, whichcannot be simulated by the SPS model.To facilitate the analysis, Fig. 5 gives the evolution curvesof the probability of the incident mode keeping its own formmode invariant with relative beam width for different OAMat K = ν = (cid:104) l · V r (cid:105) ω (5) where (cid:104)·(cid:105) is the ensemble average, vector l stands for the columnvector composed of + − V r denotes the radial distance between each vortex and opticalaxis. Our vortex detection algorithm is a modified combinationfrom [32–35].It can be seen from Fig. 5 that for different modes of incidentoptical fields under specific turbulent conditions, vortex split-ting occurs in the beam at the early stages of propagation (forthe incident light with l =
1, turbulence steers vortex off beamaxis as well as causes vortex pairs regeneration). Additionally,the transmission-induced complex effects become progressivelymore pronounced as W increases. This behavior can be seenmore clearly in the curves plotted as the variance of t insteadof W . With the increase of the vortex splitting ratio ( ν ≥ l = ν ≥ l = ν ≥ l = esearch Article Journal of the Optical Society of America A 6 Fig. 6.
OAM spectra, averaged over 300 realizations of turbulence and phase distribution of OAM-carrying beam for a singlerealization of turbulence with azimuthal index (a), (d) l =
1; (b), (e) l =
2; (c), (f) l = ν = ν = ν = ν = ν = ν = l = l = − l =
1, which will inevitably causethe probabilistic evolution of the mode crosstalk to deviate fromthe results of the Paterson C theory.Therefore, the emergence of the deviation scale is actuallythe effect of the continuous accumulation of vortex splitting,for the short-range weak scintillation case while the conditionsfor the emergence of the deviation conditions are not satisfied,even with the generation of vortex splitting arising from theintensity perturbation, the mode crosstalk evolution probabilityat this moment is still consistent with the results given by theSPS model as well as the average OAM also remains basicallyunchanged. As shown in Fig. 6, we verify the OAM spectraldistribution of the received optical field after 300-iterations en-semble average under the circumstance of the weak scintillationwith short-range (not exceeding the threshold) and with long-range (passing the threshold). By comparing the three plotsbetween the upper and lower, we can see that when the incidentmode is propagated over a short distance as well as the vortexsplitting ratio did not exceed the threshold, the OAM spectrumdistribution of the received optical field almost coincides withtheoretical prediction provided by Paterson C. After a certaindistance of transmission while passing the threshold, the numer-ical simulation results deviate from the theory, at this point, thecumulative compound effect on the evolution of optical fieldmode becomes more destructive, which indicates that it is notscientific for us to use the phase perturbation approximationto simulate the compound effect of turbulence even though thefree-space channel is at weak scintillation scale. In other words,such circumstances also happen to be unpredictable by the scin- tillation index and need to be measured with the help of thevortex splitting ratio.
5. CONCLUSIONS
The introduction of the IPE equation with Roux can give theevolution rule of a single photon or multiple closely spaced pho-tons perturbed while traveling through atmospheric turbulencein the case of all scintillations. In this paper, we first reexaminethe minimum parameter dependence determined the evolutionof OAM crosstalk in turbulent link while considering the radialmode scrambling. Then, we present a qualitative understandingof the deviation scale under arbitrary turbulent strengths bycomparing the difference between the numerical value and SPStheoretical prediction. Finally, by calculating the vortex splittingratio, the endogenous physical properties of the deviation scaleare elaborately explained in this contribution. It found thatwhen the Rytov variance σ R < < σ R < t <
1, the evolution of OAM crosstalk probability almostcoincides with the theoretical prediction. Beyond the Rayleighrange ( t > ) , the deviation scale starts to emerge and graduallybecomes widely divergent. Moreover, by detecting the splitvortices caused and contributed by intensity fluctuation, theinfluence of t on the emergence of the deviation scale can bemeasured by the vortex splitting ratio, when ν passed thespecific threshold for different incident modes, e.g. ν = l =
2, the turbulent spatial accumulation effect makes theindividual vortices in the incident mode gradually separated,meanwhile, the probability of finding a single vortex in thereceived mode increased, which makes the simulation curvedeviate from the theoretical prediction in the weak scintillationcase. Therefore, for the vortex beams that are perturbed whilepropagating through the free-space channel, it is undesirable touse the phase perturbation approximation for simulating theturbulence effect at the weak scintillation scale with long-range.
Funding.
Anhui Provincial Natural Science Foundation (GrantNo. 1908085QA37); National Natural Science Foundation of esearch Article Journal of the Optical Society of America A 7
China (Grant No. 11904369); State Key Laboratory of PulsedPower Laser Technology Supported by Open Research Fund ofState Key Laboratory of Pulsed Power Laser Technology (GrantNo. 2019ZR07).
Acknowledgment.
The authors would like to thank Dr YichongRen and Dr Ruizhong Rao for their careful reading andinsightful suggestments of our manuscript.
Disclosures.
The authors declare no conflicts of interest.