Theoretical analysis of the optical rotational Doppler effect under atmospheric turbulence by mode decomposition

Sheng-Jie Ma(马圣杰), Shi-Long Xu(徐世龙),†, Xiao Dong(董骁), Xin-Yuan Zhang(张鑫源),You-Long Chen(陈友龙), and Yi-Hua Hu(胡以华),‡

1State Key Laboratory of Pulsed Power Laser Technology,National University of Defense Technology,Hefei 230037,China

2Key Laboratory of Electronic Restriction of Anhui Province,National University of Defense Technology,Hefei 230037,China

Keywords: optical rotational Doppler effect, atmospheric turbulence, vortex beam, mode decomposition,mode crosstalk

1.Introduction

The well-known Doppler frequency shift can be expressed as Δf=2vfc-1, which shows that when there is a linear relative motion between the source and the observer, a frequency shift will be generated,and this phenomenon is also called the linear Doppler effect (LDE).The frequency shift varies with the relative velocity between the source and the observer, and it is extensively used in Doppler velocity detection along the line of sight.[1,2]When a beam with a helical phase structure (such as a vortex beam) illuminates a rotating object with an optically rough surface, another type of Doppler frequency shift will occur, namely the rotational Doppler effect (RDE).[3-7]In 1992, Allenet al.discovered the vortex beam with a helical phase factor exp(ilφ) carries orbital angular momentum (OAM),[8]whereφis the azimuthal angle,lis the topological charge (TC), andlis also called the OAM mode index.Since then,the vortex beam has been widely researched in optical communication,[9-11]optical manipulation,[12,13]and other fields.In recent years,vortex beams have also been widely used in rotational velocity detection,angular velocity detection,locating the center of rotating objects based on RDE.[5,6,14,15]

In 2013,Laveryet al.designed a rotational velocity measurement scheme based on RDE by collecting and analyzing the OAM beam scattered from a spinning object with an optically rough surface.[4]They discovered that the rotational Doppler frequency shift of a rotating object illuminated by an OAM-carrying beam can be expressed by Δf=lΩ/2π,which showed that the frequency shift is only related to the rotational velocity of the spinning object and the OAM mode index.Since then,many kinds of research on rotational velocity detection based on RDE have been published.[16-18]Qiuet al.analyzed the RDE at the oblique light incidence and formulated the quantitative relation between the Doppler frequency shift and the tilt angle.[16]In the same year, they experimentally analyzed the influence of lateral misalignment on RDE.They concluded that the Doppler signal would be spectrabroadened when the beam axis of a probe light is not coincide with the rotation axis of the spinning object.[17]Based on this research, Zhanget al.analyzed the rotation velocity detection with orbital angular momentum light spot completely deviated from the rotation center.[18]However, most existing studies[14-18]are based on ideal conditions,and the influence of atmospheric disturbance on a vortex beam has not been considered.In Ref.[19], Zhanget al.established a 120 m long free-space link and experimentally analyzed the influence of atmospheric turbulence(AT)on the RDE.But they did not explain how AT affects the rotational Doppler frequency shift.In fact,especially for the free-space optical link,the propagation of vortex beams will be influenced by many factors, such as atmospheric turbulence, atmospheric scattering, aerosol, and so on.In this paper, we mainly analyze the influence of AT on vortex beams.The AT will lead to random variations in the refractive index, and the wavefront of the beam will be inevitably distorted, which will lead to a series of influences such as inter-symbol interference,signal power reduction,and so on.[20]The impact of AT on optical communication systems has been studied by many researchers.[21-23]However,the influence of AT on RDE has not been analyzed in the existing research, which is a non-ignorable part of RDE for practical applications.

In this paper,we theoretically analyze the influence of AT on RDE by mode decomposition and deduce the relationship between the rotational Doppler signal frequency spectrum and the AT intensity.Theoretical analysis shows that AT will lead to the expansion of the rotational Doppler signal frequency spectrum to adjacent modes.And whenC2n ≤5×10-15m-2/3and 2z ≤2 km,the rotational Doppler signal frequency spectrum widthdand the spiral spectrum widthd0satisfies the relationship ofd=2d0-1.Finally,we analyze the influence of mode crosstalk on RDE,and the result indicates that mode crosstalk will lead to an asymmetric distribution of the rotational Doppler signal frequency spectrum.The research in this paper may help us better understand the mechanism of RDE and provide theoretical guidance for rotational object detection and OAM beam measurement.

2.Theoretical analysis

When a beam with a helical phase structure, such as the Laguerre-Gaussian(LG)beam,the Bessel beam,or the Airy-Gaussian beam, illuminates the surface of a rotating object,the rotational Doppler effect will occur.[16,24,25]Taking the LG beam as an example,the light field of the LG beam propagating along thez-axis can be expressed as

In Ref.[26], Fanget al.concluded that the RDE shared a common origin with LDE and the formula of rotational Doppler frequency shift could be derived from the linear Doppler frequency shift.The well-known linear Doppler frequency shift can be expressed as

wheref0is the frequency of the source light,vis the relative velocity between the source and the observer,andθis the angle between the light propagation direction and the relative velocity vector direction.Whenθ=π/2,i.e.,the light propagation direction is perpendicular to the velocity vector direction, the linear Doppler shift equals 0 according to Eq.(2),which implies the LDE is not suitable to measure the spinning velocity.As for a vortex beam with a helical wavefront,there is an oblique angle between the Poynting vector and the beam propagation direction, which can be used for the application of RDE.As shown in Fig.1(a),the angle between light propagation direction and Poynting vector can be expressed as sinα=lλ/(2πr).[4]Under this circumstance,α+θ=π/2 and cosθ=sinα=lλ/(2πr).Thus,Eq.(2)can be rewritten as

whereΩis the rotational velocity of the spinning object with an optically rough surface.Equation(3)shows the basic form of the rotational Doppler frequency shift.Unlike the LDE,the frequency shift based on RDE is only related to the rotational velocityΩand the OAM mode indexl.And it has nothing to do with the frequency of the source,which has been experimentally proved in Ref.[27].In most existing experimental schemes,the light scattered from a spinning object with an optically rough surface is collected by a photodetector(PD),and the frequency shift spectrum can be obtained through real-time Fourier transform by an oscilloscope.[14-18]However, the accuracy of the experimental measurements depends on many factors.For example,when the power of the scattered light is too small, the PD may not receive any signal,[3,16]and when the reflectivity of the spinning object surface is not homogeneous,there will be both amplitude and phase modulation for the incident beam,which will reduce the Doppler shift.[28]As shown in Fig.1(b),assuming that the OAM mode index of the vortex beam scattered from a spinning object ism, then the rotational Doppler frequency shift can be rewritten as

whereβis the scattering angle.According to Eq.(4), when an OAM-carrying beam illuminates a spinning object with an optically rough surface,the frequency shift is related to the difference of the mode index between the incident and scattered beam and the rotating velocity.

On this basis, we add the influencing factor of AT.As is shown in Fig.2(a), the vortex beam first propagates in the AT, and after being scattered by the rotating object, the scattered beam passes through AT again and finally received by the detector on the receiving end.Therefore, it is necessary to analyze the three processes of “propagation-scatteringpropagation” of the vortex beam.Figure 2(b) shows the equivalent model of two-way propagation.When the vortex beam passes through AT, its mode index will diffuse to adjacent modes and we assume that the mode index of OAMlwill expand toL, and then the mode indexes of the scattered vortex beam will be further diffused toL＇, andL＇contains{l1,l2,...,ln}.Then, the rotational Doppler frequency shift can be expressed as

The superimposed LG beam with two opposite TCs is widely used in RDE experiments.When propagating in AT,the mode index of the OAMl,-lwill expand toL＇and-L＇.Then the rotational Doppler frequency shift for thelpart can be expressed as

Similarly, the rotational Doppler frequency shift for the-lpart can be expressed as

whereliis inL＇andljis in-L＇, representing the expanded mode indexes of the OAMl,-lbeam.Thus, the rotational Doppler frequency shift can be expressed as

At this point, the frequency shift no longer has anything to do with the scattered beam mode indexm, but only with the extended modesL＇,-L＇, and the spinning velocityΩ.Compared with the single LG beam, the superimposed LG beam no longer needs to consider the scattered beam mode index,which also avoids the experimental errors caused by the inaccurate measurement of the scattered beam mode indexm.When the rotating velocityΩis a constant,the frequency shift is only related to the extended mode indexes, and how the OAM mode index will expand is related to AT intensity.

Fig.1.(a)Schematic diagram of the incident OAM beam Poynting vector and object rotation axis.(b) Schematic diagram of the incident and scattered OAM beam mode index.

Fig.2.(a)The two-way propagation model of vortex beam in AT.(b)The equivalent model of two-way transmission.

In the following, we will analyze the influence of AT on the OAM beam mode index by mode decomposition.The optical fieldu(r,θ,z)of the LG beam can be expanded by a helical spectrum function exp(imθ)as[29]

wheremis an integer number andam(r,z) is the spiral spectrum coefficient,which can be expressed as

Then the square of modulusam(r,z)can be expressed as

For AT,〈exp[φ(r,θ1,z)+φ∗(r,θ2,z)]〉can be expressed under the Rytov approximation as

whereDφ(·) is the phase structure function,r1,r2represent the vector directions of (r,θ1),(r,θ2), respectively,r0=(0.545C2nk2z)-3/5is the coherence length of AT,andC2nis the refractive index structure constant, which is usually used to measure the AT intensity.

To evaluate Eq.(11),we introduce the Bessel integral formula

whereIn(·)is the modified Bessel function of integer ordern.The energy content of different mode indexes can be obtained by radial integration of|am(r,z)|2

whereRis the radius of the receiver aperture.Then themth spiral spectrum components can be determined by the expression

In the absence of AT, the OAM beam with the mode indexlhas the normalized powerPm= 1 form=landPm= 0 form/=l.In addition, the value ofPmremains unchanged with the increase of the propagation distance.When the OAM beam with mode indexlpropagates in AT, the value ofPmform=lwill decrease and form/=lwill increase due to the random phase perturbations caused by AT.According to Eqs.(14) and (15), it is not difficult to find that mode index diffusion caused by AT has a symmetrical distribution aboutl, that is to say, the value ofPl+ΔlequalsPl-Δl.In this case, we can reasonably assume that the mode indexes of OAMlwill expand to{l-2,l-1,l,l+1,l+2}when propagating in AT.And the mode indexes of OAM-lwill expand to{-l-2,-l-1,-l,-l+1,-l+2}similarly.Under this circumstance, the rotational Doppler signal frequency shift will expand from{2l}to{2l-4,2l-3,2l-2,2l-1,2l,2l+1,2l+2,2l+3,2l+4}inΩ/2πunits,and the power of each mode is determined by the corresponding energy weight according to Eq.(15).

According to the above theoretical analysis,we can conclude that the random phase perturbations caused by AT will lead to the expansion of the rotational Doppler signal frequency spectrum.The rotational Doppler signal frequency spectrum widthdis dependent on the spiral spectrum widthd0under the influence of AT,and the relationship between them can be expressed asd=2d0-1.

3.Simulation results and analysis

In Eq.(15),mcan take the value of any integer,that is to say, the spiral spectrum can be expanded infinitely under the influence of AT.WhenPmtakes a very small value, we can consider there is no expansion anymore.In order to ensure the integrity of the spiral spectrum,we should first ensure a certain range of topological charges aroundlbased on Eq.(14).Without loss of generality,we set the range of topological charges froml-10 tol+10 in this paper.

3.1.The influence of AT intensity on RDE

We first analyze the expansion of the spiral spectrum under different AT intensities.In the simulation,we set the initial parameters of the LG beam withλ=632.8 nm,ω0=0.01 m,l= 12 and±12.Figures 3(a) and 3(b) show the beam intensity distribution and the spiral spectrum of the single LG beam withl=12 and the superimposed LG beam withl=±12 and the propagation distance is 2z=1 km.Without the influence of AT, the light intensity of the single LG beam withl=12 presents a circular distribution, and the superimposed LG beam withl=±12 has a petal-shaped intensity distribution, and the number of petals is 2l.Under the influence of AT, the light intensity of the LG beam will be distorted, and the perfect ring-shaped light intensity structure begins to be deformed.As the intensity of AT increases,the dispersion of the light intensity also increases.WhenC2n=1×10-14m-2/3,the perfect ring-shaped light intensity structure is no longer visible.It is not difficult to find that the superimposed LG beam withl=±12 will not affect the accuracy of the spiral spectrum, and the superimposed LG beam shows exactly the same spiral spectrum symmetrical about 0.When there is no AT,the spiral spectrum of the single LG beam withl=12 is only distributed atl=12,and the superimposed LG beam withl=±12 shows a symmetrical distribution about 0 atl=12 andl=-12.WhenC2n=1×10-16m-2/3, the spiral spectrum gradually expands tol=±11 and±13.However,as the values ofPl=±11andPl=±13are very small,we can ignore the expansion.With the increase of AT intensity, the spiral spectrum gradually expands to the adjacent modes.WhenC2n=1×10-15m-2/3,it expands tol=±11 andl=±13 and whenC2n=1×10-14m-2/3,the spiral spectrum contains 18 modes{±8,±9,±10,±11,±12,±13,±14,±15,±16}.Meanwhile,the normalized power ofl=±12 decreases rapidly, and its value is less than 0.4.Under this circumstance,the expansion of the spiral spectrum is very serious.

Fig.3.Distribution of beam intensity and spiral spectrum under AT when propagation distance 2z=1 km.(a)The single LG beam with l=12 and(b)the superimposed LG beam with l=±12.

Fig.4.Distribution of rotational Doppler signal frequency spectrum under different AT intensity.

The rotational Doppler signal frequency spectrum under different AT intensities is shown in Fig.4.WhenC2n= 1×10-15m-2/3, the spiral spectrum consists of six modes, including{±11,±12,±13}.According to Eq.(8),two modes ofl= 12 andl=-12 will generate the frequency shift Δf= 24Ω/2πandl= 12 andl=-13 will generate the frequency shift Δf= 25Ω/2π.Similarly,the rotational Doppler signal frequency spectrum will include{22,23,24,25,26}inΩ/2πunits.WhenC2n= 1×10-14m-2/3,the spiral spectrum consists of 18 modes,including{±8,±9,±10,±11,±12,±13,±14,±15,±16}and the Doppler signal frequency shift spectrum should theoretically contain 17 modes{16,17,...,32}inΩ/2πunits.However,the width of the rotational Doppler signal frequency spectrum is less than the theoretical width in fact.As shown in Fig.4(d),there are only 11 modes{19,20,...,29}inΩ/2πunits.The reason may be that when spiral spectrum mode diffusion occurs, the modes adjacent tolhave greater normalized power,and the normalized power of the remote modes will be further reduced or even be neglected in the calculation of the rotational Doppler frequency shift spectrum.Under this circumstance, the relationship between the rotational Doppler signal frequency spectrum widthdand the spiral spectrum widthd0no longer satisfies the relationship ofd=2d0-1 butd ＜2d0-1.

The relationship curves ofdandd0under different AT intensities are shown in Fig.5.WhenC2n ≤5×10-15m-2/3,the values ofd0are 1, 1, 1, 3, and 5, and the corresponding values ofdare 1,1,1,5,and 9,respectively,and the values ofdandd0satisfy the relationshipd=2d0-1.When the value ofC2nincreases to 1×10-14m-2/3,d=10 andd0=9 and they no longer satisfy the relationshipd=2d0-1 butd ＜2d0-1.Therefore,we can conclude that whenC2ntakes a small value,i.e., under weak AT,the spiral spectrum expansion caused by AT will lead to the expansion of the rotational Doppler signal frequency spectrum,and the rotational Doppler signal frequency spectrum widthdsatisfies the relationshipd=2d0-1.WhenC2ntakes a large value, the relationship between them changes tod ＜2d0-1.

Fig.5.Spectrum width of the spiral spectrum and rotational Doppler signal frequency spectrum under different AT intensities.

3.2.The influence of propagation distance on RDE

In addition,we setC2n=1×10-15m-2/3as a fixed value to analyze the influence of the propagation distance (2z) on the rotational Doppler signal frequency spectrum.Figure 6(a)shows the normalized power atl=12 of the single LG beam andl=±12 of the superimposed LG beam at different propagation distances.The normalized power of the single LG beam and the superimposed LG beam are 1 and 0.5 at the initial position, respectively.When the propagation distance increases to 2z=6 km, the normalized power of the single LG beam and the superimposed LG beam reduces to 0.536 and 0.268,respectively.And under the same propagation distance,the normalized power of the superimposed LG beam is always half of the single LG beam, which is consistent with theoretical analysis.Figures 6(b) and 6(c) show the distribution of the spiral spectrum and the Doppler signal frequency spectrum under different propagation distances.When the propagation distance of the LG beam is less than 2z=2 km,the spiral spectrum only spreads froml=±12 tol=±11 and±13.And the corresponding rotational Doppler signal frequency spectrum includes 5 modes{22,23,24,25,26}inΩ/2πunits.When 2z= 3 km and 4 km, the spiral spectrum is further expanded to{±10,±11,±12,±13,±14},and the rotational Doppler signal frequency spectrum contains 7 modes{21,22,23,24,25,26,27}inΩ/2πunits.Whenz= 6 km, the spiral spectrum extends to 14 modes{±9,±10,±11,±12,±13,±14,±15},and the corresponding rotational Doppler signal frequency spectrum also contains 9 modes{20,21,22,23,24,25,26,27,28}inΩ/2πunits.

According to the above analysis, when the AT intensity remains unchanged, the increase of propagation distance will also lead to rotational Doppler signal frequency spectrum expansion.It is not difficult to find that in the case of a short distance propagation(2z ＜2 km),the rotational Doppler signal frequency spectrum widthdand the spiral spectrum widthd0also satisfy the relationshipd=2d0-1,andd ＜2d0-1 in the case of a long distance propagation.

Fig.6.(a)The normalized power at l=12 of the single LG beam with and l=±12 of the superimposed LG beam under different propagation distances;(b)and(c)are the spiral spectrum and rotational Doppler signal frequency spectrum,respectively.

3.3.The influence of mode crosstalk on RDE

Fig.7.(a)Spiral spectrum with different mode indexes and AT intensities.(b)Rotational Doppler signal frequency spectrum corresponding to panel(a).

Table 1.Normalized power of different modes with and without crosstalk.

In the above analysis, we set the value of TC tol=12 of the single LG beam andl=±12 of the superimposed LG beam.The reason why we choose a large value of TC is to avoid the crosstalk between different mode indexes caused by AT.[30]The crosstalk will ultimately lead to an increased bit error in the receiving end and a decreased capacity for the optical communication system.[31]Finally, we will analyze the influence of mode crosstalk on RDE.According to the spiral spectrum in Fig.3(b), even whenC2n=1×10-14m-2/3, we can still clearly see two peaks of the superimposed LG beam atl=12 andl=-12.The minimum diffusion mode ofl=12 isl=8,and the maximum diffusion mode ofl=-12 isl=-8,so there is no intersection in the spiral spectrum, and the rotational Doppler signal frequency spectrum also presents one peak spectrum at 2linΩ/2πunits.If the TC of the superimposed LG beam takes a small value,crosstalk between different modes is likely to occur.Figure 7(a)shows the spiral spectrum with different values oflandC2n.In order to better verify this conjecture, we letC2ntake some large values: 5×10-15,1×10-14, 5×10-14, 1×10-13m-2/3.Forl=±12, mode crosstalk only occurs whenC2n=1×10-13m-2/3.However,the mode crosstalk does not have a significant influence on the rotational Doppler signal frequency spectrum,and the two peak spectrums atl=12 andl=-12 can still be clearly identified.Forl=±8,mode crosstalk begins to appear whenC2n=5×10-14m-2/3and the normalized power atl=0 is close to half of the maximum value whenC2n=1×10-13m-2/3,which has a certain influence on the whole signal spiral spectrum.Forl=±4,whenC2n=5×10-14m-2/3,the normalized power atl=0 is 0.05, which is close to the normalized peak power.WhenC2n=1×10-13m-2/3,the normalized power atl=0 is already bigger than that atl=±4.What is worse, the original two peaks spectrum degenerated into one peak spectrum atl=0.At this time,the crosstalk between different modes was very serious, and the value oflcould not even be intuitively determined by the spiral spectrum.According to the above analysis, mode crosstalk may change the structure of the spiral spectrum of the superimposed LG beam with±l,especially forlwith a small value,and the normalized power adjacent tol= 0 will increase, which will even cause the original two peaks spectrum to degenerate into a one peak spectrum.

The corresponding rotational Doppler signal frequency spectra are shown in Fig.7(b).When there is no mode crosstalk,the rotational Doppler signal frequency spectrum is symmetrically distributed about 2linΩ/2πunits.And when mode crosstalk occurs,the symmetrical distribution of the rotational Doppler signal frequency spectrum about 2linΩ/2πunits is destroyed.Table 1 shows the normalized power of six modes adjacent to 2linΩ/2πunits under different conditions.When there is no crosstalk, the normalized power of 2l-1 is the same as 2l+1 inΩ/2πunits and the rotational Doppler signal frequency spectrum is still symmetrically distributed about 2linΩ/2πunits.WhenC2n=1×10-14m-2/3,l=±4, crosstalk occurs, and the normalized powers of{2l-1,2l-2,2l-3}inΩ/2πunits are 0.107,0.100,0.080 and the normalized powers of{2l+1,2l+2,2l+3}inΩ/2πunits are 0.104,0.093,0.078, respectively.On this occasion,they are no longer symmetrically distributed about 2linΩ/2πunits.And the same phenomenon also occurs whenC2n=1×10-13m-2/3,l=±4 andC2n=1×10-13m-2/3,l=±8.The reason for this may be that when mode crosstalk occurs,the normalized power of the modes adjacent tol=0 is increased,and the spiral spectrum is no longer symmetric about±l.Consequently, the rotational Doppler signal frequency spectrum shows an asymmetric distribution about 2linΩ/2πunits.

4.Conclusion and perspectives

In conclusion, we theoretically investigate the influence of AT on the rotational Doppler effect by modal expansion and deduce the formula of rotational Doppler frequency shift under the influence of AT.We find that AT will lead to the expansion of the rotational Doppler signal frequency shift spectrum from 2linΩ/2πunits to adjacent modes,and the diffusion will increase as the AT intensity and propagation distance increase.Moreover, under strong turbulence, mode crosstalk of different modes will occur, which will lead to an asymmetric distribution of the rotational Doppler signal frequency spectrum.Our theoretical analysis provides a new supplement for RDE and may promote the development of atmospheric detection for practical applications.

Acknowledgments

Project supported by the Research Plan Project of the National University of Defense Technology(Grant No.ZK18-01-02),the National Natural Science Foundation of China(Grant No.61871389), the State Key Laboratory of Pulsed Power Laser Technology(Grant No.KY21C604),and the Postgraduate Scientific Research Innovation Project of Hunan Province(Grant Nos.CX20220007 and CX20230024).