Phenomenology of complex structured light in turbulent air

Xuemei Gu, 2, ∗ Lijun Chen, † and Mario Krenn 3, ‡ State Key Laboratory for Novel Software Technology, Nanjing University, 163 Xianlin Avenue, Qixia District, 210023, Nanjing City, China. Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria. Vienna Center for Quantum Science & Technology (VCQ), Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria. (Dated: January 23, 2020)

modes in laboratories, researchers have exploited heat pipes [18,19], random phase screens generated by spatial light modulators (SLMs) [20][21][22][23], static diffractive plates [24] and rotating random phase plates [25]. Those efforts were based on models of the turbulent atmosphere that date back to the Kolmogorov's seminal work from 1941 [26][27][28], and further extension and advances of it [29][30][31]. All of these efforts have in common that it was not clear at the time whether realistic atmospheric turbulence interacts with spatial modes of light as the models predict. The reason was simply a lack of experimental tests in real conditions.
Since 2014, the first outdoor experiments with spatial modes were conducted, with long-distance transmissions up to 143 km [32,33], with high-speed data rates up to 400Gbit/sec [34,35], and in the quantum regime with entangled photons [33] and for quantum communication [36]. These results establish the feasibility of long-distance transmission of spatial modes of light but left open the question about the predictive power of current models.
It was only in 2017 when Lavery et al. performed an experiment transmitting spatially modulated light in an urban environment and compared their results with available theoretical models. Unexpectedly, they found that current atmospheric turbulence models do not adequately predict their experimental observations [37,38]. As a consequence, they put forward an adapted method to describe spatial modes propagating in atmospherical turbulence, which can describe their experimental observations.
Here, motivated by this unusual conflict between theory and experiment, we ask what experimentally observable predictions of Lavery et al.'s model are. is equivalent to a LG mode LG n, , where p = 2n + | | and m = | |. For example, the LG1,2 mode can be described as a IG4,2, in the limit case of → 0. The upper rows describe the intensity distributions and the lower rows show the transverse phase distributions. A prominent feature of IG modes is the vortex splitting in the phase pattern, leading to multiple intensity nulls in the intensity which are controlled by .
Surprisingly, we find the prediction that the atmosphere acts basis-dependent. That means information transmitted in one basis (for instance, the famous Laguerre-Gauss (LG) basis carrying orbital angular momentum) is stronger degraded as if the information would have been encoded in another complete basis (such as the Hermite-Gauss (HG) basis). This is unexpected, as those bases span the same space of modes, and each element of the first basis can be seen as a superposition of elements of the other basis. We find this effect by studying a continuous space of states, the so-called Ince-Gauss (IG) modes [39], which have both the LG as well as the HG basis as a special case. The predicted effect can be up to 7% of the total transmission quality; thus it should be observable in already existing communication links. If such an effect cannot be observed, it would falsify the currently best model to describe spatially modulated light in the atmosphere. If the effect is indeed physical, it could be used to improve communication channels, and furthermore, indicates a novel technique to measure atmospheric properties.
The article is structured in the following way. Details about spatial modes of light are given in Sec. 2. In Sec. 3, we describe the atmospheric turbulence model used for our investigation. In Sec. 4, the numerical results and discussions are shown to illustrate the effect found in our simulation, and we explain a feasible experiment to test this prediction. The conclusions of the paper are given in Sec. 5.

II. COMPLEX SPATIAL MODES OF LIGHT
The well-known solutions to the paraxial wave equation consist of the HG and LG beams, which are derived from cartesian and circular cylindrical coordinates, respectively [40]. They both provide in principle an infinite state space and form a complete orthogonal basis, such that one can describe HG states in terms of LG modes and vice versa [41,42]. The HG modes are denoted as HG nx,ny with indices n x and n y and LG modes are described as LG n, with orbital angular momentum (OAM) index [43] and radial number n [44][45][46].
Besides these two famous state families, there is another third family of complete mode basis -IG modes, which were first introduced by Bandres and Gutiérrez-Vega [39]. These modes are exact and orthogonal solutions to the paraxial wave equation in elliptic coordinates, and of which the even IG modes are described as There u and v describe the two-dimensional elliptic coordinates. A continuous parameter describes the ellipticity and the superscript e refers to even modes. In the limit case of → 0, u and v correspond to the radial and angular coordinates of the circular cylindrical coordinates system, respectively. (p, m) ∈ N are mode number. C IG is a normalization constant and C m p (·, ) represent the even Ince polynomials [47,48]. w z is the beam radius at position z and w 0 is the beam waist at the focus z = 0. z R is the Rayleigh range, R z is the radius of curvature, λ is the wavelength and k = 2π/λ is the wave number. ϕ g = arctan(z/z R ) denotes the Gouy phase and the mode order is M = p.
We can obtain the odd IG modes IG o p,m, (u, v, z) by replacing the C IG and C m p (·, ) with S IG and S m p (·, ), which correspond to a normalization constant and the odd Ince polynomials respectively.
The helical IG modes (which, for simplicity, we call IG modes for the rest of the paper) are described as superpositions of even and odd IG modes [49,50], which are given by When the elliptic coordinates tend to the circular cylindrical coordinates, namely → 0, the IG modes Figure 2. A schematic illustration of spatial modes of light propagating through atmospheric turbulence. Here we transmit a IG mode IG7,3,2 (λ = 809nm, w0 = 25mm ) over a 1.5 km turbulent free-space link. The intensity distributions of the IG mode IG7,3,2 in the transmitted and received planes are shown. We use the empirical technique which has been experimentally demonstrated by Lavery et al. [37], to model the overall turbulent link. There the total effect of atmospheric turbulence can be represented as the approximately accumulated influence of many weakly perturbing planes. Each plane with a random phase screen stands for the turbulence along a propagation path of 10 m, and the turbulent strength of each plane is described by the Fried parameter r0 plane . In general, larger r0 plane defines weaker turbulence and in this example the number is r0 plane = 1 m (corresponding to r0 ≈ 0.05 m).
will be transferred into LG modes. In this case, the indices of states LG n, and IG p,m, are related as: | | = m and n = (p − m)/2. Additionally, when the elliptic coordinates tend to the cartesian coordinates, namely → ∞, the IG modes will be transferred into helical Hermite-Gaussian (HG) modes [49,51,52]. In this case, the indices of modes HG nx,ny and IG p,m, are related as: n x = m and n y = (p − m). This puts the IG modes in a special position between LG and HG, and therefore makes them an ideal workhorse for investigating basis-dependent effects.
For a fixed ellipticity , the IG modes IG p,m, with orthogonal mode indices (p, m) ∈ N form a complete basis. Changing the value of the ellipticity gives another complete orthogonal basis. In Fig. 1, we show the intensity and phase distributions of different IG modes IG p,m, varying their ellipticity .

III. TURBULENCE MODEL
The inhomogeneity and anisotropicity in the temperature and pressure of atmosphere result in random fluctuations of the refractive index along the propagation path of light beam [17]. Those variations of refractive index introduce the distortions of the spatially structured light fields and increase intermodal crosstalk, which dramatically affects the quality of spatial modes over long-distance link [53]. Thus understanding the turbulent behavior of the atmosphere is very crucial.
In 1941, the Russian mathematician Andrey Kolmogorov published three seminal articles, which established the foundations of statistical turbulence the-ory [26][27][28]. He found that the random statistical behavior of turbulence can be described by refractive index power density spectrum 1 , which directly relates to the phase fluctuations of light along propagation path and later is represented by the phase power density spectrum Φ mvK ϕ (κ) [17]. In our study, we apply the modified von Kármán spectrum [31], which avoids the singularity that represents energy per unit volume and becomes unbounded as the eddy size increases in Kolmogorov spectrum model. Therefore it is numerically more stable. The turbulent phase screens are generated using the modified von Kármán spectrum, which is described as [54,55] Φ mvK There r 0 is the Fried parameter [56], which describes the strength of the atmospheric turbulence along the propagation distance. Smaller r 0 indicates stronger turbulence. κ is the angular spatial frequency. α, κ m and κ 0 are constant model-dependent parameters [17,54,55] (for details about the numerical propagation and turbulence model, see Supple- In order to simulate the atmospheric turbulence along a long propagation distance, we adopt a recent empirically tested modeling technique [37,38]. In the model shown in Fig. 2, the atmosphere along 1.5 km link is split into 150 planes of weakly turbulence separated by 10 m. The total atmospheric turbulence r 0 is approximately the accumulative effect of the turbulence of each plane r 0 plane and their connection is described in Supplement 1.

IV. RESULTS AND DISCUSSION
What is the quality of spatial modes of light propagating through atmospheric turbulence over longdistance? We start with transmitting a IG mode IG 4,0, over a 1.5 km turbulent channel. We call radial-like modes for IG states IG p,m, when they are equivalent to radial modes LG n,0 in the limit case of → 0. Analogously, we call OAM-like modes for IG states IG p,m, when they are equivalent to OAM modes LG 0, in the limit case of → 0. Therefore, the IG mode IG 4,0, used in our simulation corresponds to a radial-like mode, with two intensity rings with = 0. Here a question naturally arises: What role does the ellipticity play on the transmission quality of IG modes under different turbulent conditions? For simplicity, we use two different ellipticity → 0 and = 4 and propagate the IG modes through the atmosphere of different turbulent strength r 0 . The transmission fidelity describes the "closeness" of the received propagated light field |Ψ turb and the undisturbed light field in the observed planes |Ψ vac . The fidelity is given by the squared overlapping the two light fields The average fidelity is obtained by averaging over roughly 1000 individual transmissions.
In Fig. 3 (a), we show the average fidelity of IG 4,0, under different turbulence conditions. The result clearly shows the fact that under weak turbulence (equivalent to the case of large r 0 ), the quality is better than that under strong turbulence. Interestingly, we find that the ellipticity plays an unexpected role in the transmission quality of IG modes along the turbulent path. To our surprise, there is a significant increase for the radial-like modes with a large ellipticity propagating through strong turbulence. In Fig. 3 (b), we show the fidelity difference versus various turbulent strengths. We find that there is a large fidelity difference when the turbulent strength is r 0 ≈ 0.05 m, which is a realistic turbulence condition (we use this r 0 for the rest of our simulations). With this observation, we would expect that the difference will continue to increase by enlarging the ellipticity of the radial-like mode IG 4,0, . This indicates that helical HG modes, corresponding to radial-like modes in the case of → ∞, perform better under strong turbulence than that by LG radial modes. . The propagation quality of IG modes under different turbulence conditions. a: We transmit a radiallike mode IG4,0, ( : 0, 4) over 1.5 km turbulent link of different strengths r0. Large r0 describes weak turbulence and indicates good transmission fidelity. We find that the quality of IG modes in the case of = 4 is significantly better than that in the case of → 0. b: We show the fidelity difference versus turbulent strength r0. There is a large difference when the turbulent strength is r0 ≈ 0.05 m. If we continue to increase r0 (which means turbulence becomes weak until there is no turbulence), the difference will decrease until zero.
The transmission quality for radial-like modes increases when we increase the ellipticity . We would expect that this increase is compensated by OAMlike modes whose quality decreases when we increase . However, the results in Fig.4 show that the transmission quality of all other modes with order N = M stays constant (within significant uncertainty). This means that the average propagation quality of order M = 4 increases when we change to a basis with a large ellipticity 3 . This means that the atmosphere influences bases with small in a stronger way than bases with large . This is a so far unknown basisdependent, physical effect of atmospheric turbulence, and its uncovering is the main results of our paper. Furthermore, we find that this effect consistently exists for order M = 2 up to M = 6 (which contains 3   and 7 modes, respectively) and the results are shown in Table. IX 4 . An interesting insight into this effect is a cross-talk matrix of the radial-like mode IG 4,0, with small and large ellipticity. Indeed, in Fig. 5 we observe that small ellipticity leads to larger cross-talk with other modes of this basis, whereas larger ellipticity reduces the cross talk. The mathematical description of this effect would be exceptionally interesting, but is out of the scope of this manuscript.
There is now one important question that remains: Is this a physical effect or an artifact of the currently most trusted model for the propagation of complex spatial modes? This question can only be solved by experiments. An experimental investigation would require a long-distance outdoor link for transmitting spatial modes of light, which already exists in several places worldwide. At the sender, one requires a high-quality construction of complex spatial modes,  Figure 5. The mode crosstalk matrix for radial-like modes IG4,0, propagating through turbulence. In the limit of → 0, a chosen basis with IG indices set (p, m) can be rewritten as a LG indices (n, ), in our example ∈ (-7,7) and n ∈ (0,6). Each element in the matrix stands for the fidelity between the mode after transmission of 1.5 km, and an undisturbed mode. In a vacuum, only one element would have value F=1 and everything else would be zero. a: The mode crosstalk matrix for IGp,m, with → 0. b: The mode crosstalk matrix for IGp,m, with = 4. We can see that in the case → 0 (LG modes), the fidelity is spreading over significantly more modes than for = 4. The physical reason for this phenomenon should be a target for a follow-up investigation.
for instance using the technique presented in [59]. At the receiver, the mode needs to be measured -for example by transforming it to a Gaussian mode with high quality [60] and using a Singe-Mode Fiber as a filter [61,62]. Atmospheric conditions (in particular the Fried parameter r 0 ) are stable for long enough to successfully perform measurements in the form of Fig.  4 and Fig. 5.

V. CONCLUSION
In 2017, Lavery et al. [37,38] have tested for the first time theoretical models that describe spatially modulated light propagating through the atmosphere in an experiment. They found that current models do not adequately describe the experimental results and put forward an empirically valid method -multiple random phase screens that follow Kolmogorov's statistics stacked at a very close distance from each other.
Here, we employ their empirically developed model and use it to investigate the propagation of very general spatial modes of light in realistic conditions. Surprisingly, the model predicts that the effects of the atmosphere are basis-dependent. That means information encoded in one complete basis of spatiallymodulated beams can be transmitted with different quality depending on which basis one uses. What is more, the influence of this effect can be up to 7% of the total transmission quality, which makes it suitable for experimental observation. To observe this prediction, we describe an experiment which is feasible with today's technology and communication channels that have already been established in several locations worldwide.
We envision a new scientific program, which we call the phenomenology of the standard model of complex atmospheric light propagation: Here, the goal is to distill new, empirically observable phenomena of models, and pinpoint down differences between different models that can be measured in experiments. Those contributions will allow experimental physicists to falsify models, and observe new effects in natural environments which might -apart from its pure scientific purpose -have an impact in practical questions such as classical and quantum communication, or potentially novel methods to measure properties of atmosphere and thereby weather dynamics. Our results can be seen as an inaugural contribution to this program.

A. Turbulence Strength and Model
In the presence of a turbulent atmosphere, spatial modes of light experience atmospheric refractive index variations caused by fluctuations in temperature and pressure. These atmospheric refractive index variations distort the wavefronts of the propagated light beams [17]. A measure of the strength of random fluctuations is the refractive index structure parameter C 2 n (z), which is used to quantify the strength of the atmospheric turbulence along the propagation path. Typical values of C 2 n (z) range from 10 −17 m −2/3 in weak turbulence up to 10 −13 m −2/3 in strong turbulence [17,55].
Another parameter often used to estimate the integrated strength of turbulence, especially in connection with astronomical imaging, is the Fried parameter r 0 [63]. Stronger turbulence corresponds to a smaller r 0 . For a known refractive index structure parameter C 2 n (z) along the propagation path, the Fried parameter r 0 is given by [64][65][66] There k = 2π/λ is the wavenumber and λ is optical beam wavelength (in our simulations, a wavelength of 809nm is adopted). α 1 = 0.423 is a constant number which derived in the case of the phase variance is approximately one [55,64]. The integral is taken over the overall propagation path from the transmitter to the receiver plane.
We exploit the experimentally demonstrated model from Lavery et al. to simulate atmosphere over longdistance. There the total 1.5 km turbulent link is decomposed into many short weakly perturbing planes separated by 10 m. In general, C 2 n (z) is assumed to be roughly a constant over short time intervals or propagation distance. Using Eq.4, we could approximately describe the overall strength of atmosphere r 0 as a accumulation of strength in every plane r 0 plane , which is described as The parameters used for studying the propagation of spatial modes of light under different turbulence conditions are listed in Table. II. For the purpose of numerically modeling turbulence, we use the modified von Kármán phase power spectrum Φ mvK ϕ (κ) to generate random phase screens [31,55], which is described as where κ is angular spatial frequency in rad/m. α 2 = 9.7 × 10 −3 , κ m = 5.92/l 0 and κ 0 = 2π/L 0 are constant model-dependent parameters [54,55]. L 0 and l 0 are the so-called outer and inner scale 5 , typically L 0 = 100 m and l 0 = 0.01 m [54]. The phase power spectrum Φ mvK ϕ (κ) can also be written in terms of a Fried parameter r 0 by combing Eqs. 4and 6, which is

B. Numerical Simulation and Results
In our simulation, we adopted a collimated light beam of beam waist w 0 = 25mm and wavelength λ = 809nm. All parameters used in our numerical simulation are presented in Table. III. In the paper, we have investigated the propagation quality of IG modes IG 4,0, with different ellipticity ( : 0, 4) under different turbulence conditions. The propagation fidelity is calculated by the squared overlapping the received propagated light field |Ψ tur and the undisturbed light field in the observed plane |Ψ vac , which is given by The average fidelity is computed by averaging all individual transmissions. The average fidelity of IG 4,0, through different turbulent strengths are described in Table. IV. There the fidelity is in percentage and the error is given by the standard deviation of the mean. The results show that the ellipticity plays an unexpected role in the transmission quality of IG modes along the turbulent path. To our surprise, under strong turbulence, there is a significant increase for radial-like modes IG 4,0, with a large ellipticity. In addition, there is a large fidelity difference between the radial-like modes with ellipticity → 0 and = 4 when the strength of atmospheric turbulence is r0 ≈ 0.05 m.
Then we chose a set of modes {IG 4,0, , IG 4,4, , IG 4,2, } and analyse their transmission fidelities through atmosphere of a turbulent strength r 0 ≈ 0.05 m. The results are described in Table. V, which show that, with a large ellipticity, there is a significant increase in the fidelity of radial-like modes instead of OAM-like modes. Furthermore, we investigate our finding by analysing the mode crosstalk matrix for the radial-like modes IG 4,0, with different ellipticity . In the limit of → 0, a chosen basis with IG indices set (p, m) can be rewritten as a LG indices (n, ). Thus, labeled columns represent the modes with index , meanwhile labeled rows represent the modes with index n. Each element in the matrix is given by an inner product measurement of the transmitted field |Ψ tur and each undisturbed mode (individually) from the set of basis |vac j , which is given by We show the values of the mode crosstalk matrix of IG modes IG 4,0, in Tables. VI and VII. We can see that the mode crosstalk becomes less for radial-like modes by increasing and the fidelity of the propagated mode (highlight in green) is significantly larger than that with small . In order to ensure the accuracy of the numerical results in the article, we perform a vast amount of sanity checks. It involves the analyzation of the normalization of the modes which we confirm in Table. VIII and Fig. 6, the verification of orthogonality between different LG n, modes in Fig. 7 and the decomposition and orthogonality of a LG n, mode in the HG basis in Fig. 8.
Furthermore, it involves the comparison between the numerically propagated and theoretically calculated LG, HG and IG modes in Fig. 9 and with a large grid number in Fig. 10. Also we investigate the propagation of a single higher order LG mode through turbulence and study their propagation, the intensity and their fidelity in Fig. 11.
Finally, in order to verify the mode-dependent effect described in the article is not a numerical artefact, we perform a calculation of pure LG modes comparing to pure HG modes with a even large grid number. We see those values are within the statistical uncertainty to the values in the main text and show the effect described in the article. Our chosen discrete parameter fulfills the geometrical and aliasing constraints for numerical beam propagation [54,57,58]. Therefore we are confident that the results presented in the paper are due to the model instead of the numerical inaccuracy. Table VIII. We analyse the modes space containing in the receiver plane. For that, we calculate different theoretical high-order LG modes and show which of these modes have more than 99.99% intensity within the receiver plane. We see our receiver plane can complete contain more than 400 modes in the LG basis. . We show intensity and phase distributions for some high-order modes investigated in Table. VIII. We also graphically show the results from Table. VIII in b. Yellow means these modes are contained within our receiver plane and blue means those modes have less than 99.99% of the intensity in our receiver plane. In this case, the grid number is N = 1024.  Figure 7. We investigate the quality of the numerical beam propagation without turbulence via analysing the orthogonality of the modes in the receiver plane. We transmit a LG2,1 mode without turbulence over 1.5km and calculate the fidelity between a set of 16 theoretical calculated LG modes at the receiver. We see that the expected mode have fidelity more than 99.99%. While the mode we propagated has the fidelity with all the other modes less than 10 −17 . This confirms that the propagation of the LG modes works as expected. In this case, the grid number is N = 1024.   Figure 8. We investigate again the quality of the numerical beam propagation without turbulence by decomposing a spatial mode into a different basis. Here We transmit a LG2,1 mode without turbulence over 1.5km and analyse the fidelity of the receiving beam with 49 theoretical calculated HG modes. The LG2,1 has a mode order M = 5, thus it can be seen as a sum of HG modes with mode order M = 5. As we expected, we find that the propagated mode has a high overlap with the HG modes of order M = 5 and less than 10 −29 overlapping with other modes. It confirms the LG and HG basis work correctly through numerical propagation and the orthogonality is conserved. In this case, the grid number is N = 1024. Figure 9. In order to confirm further the numerical propagation works for high-order modes, we send 36 different modes from LG, HG and IG basis without turbulence. We calculate the fidelity of the propagated modes with the theoretical modes in the receiver plane. For LG and HG modes, we find the fidelities are larger than 99.99% for all the 36 modes. For IG modes, we find the overlap is larger than 99.5% for the highest modes up to order M = 10. Thereby, we see our numerical propagation works with high quality. In this case, the grid number is N = 1024. Figure 10. We do the same sanity check with the grid number N = 2048 for the calculation in Fig. 9. For LG and HG modes, we find the fidelities are larger than 99.99% for all the 36 modes. For IG modes, we find the overlap is larger than 99.9% for the highest modes up to order M = 10. Figure 11. We propagate the LG2,1 mode with turbulence and analyse its propagation at different stages. In a, we see that the intensity and phase distribution after 6 consecutive steps of 300m propagation changes continuously. In the upper picture of b, we see the intensity of the propagated mode continuously stay above 99.99%, which confirms that we don't loss intensity within the propagation. Furthermore, we calculate the fidelity of the beam with the large mode space from Table. VIII (there we cut off the mode basis from | | larger than 77). We find the fidelity stays at a very high value overall. The fidelity is larger than 99.91% and 99.81% after 900m and 1500m propagation respectively. This confirms our propagation quality is high. In c, we show the crosstalk fidelity of the propagated mode after 300m, 900m and 1500m with 102 modes. We can see clearly how the fidelity changes and the transmitted mode propagates to other modes. Overall, this again confirms our propagation through turbulence fulfills our sanity checks. Thus we are confident about the validity our results. In this case, the grid number is N = 1024.