In this paper, an optimal mode set is proposed to maximize the receiving power for free space transmission under atmosphere turbulence with transmitter/receiver aperture size constraint. The optimal beam profiles are evaluated through eigenmode analysis of the Fredholm integral equation, which is mathematically equivalent to the eigen vector analysis of an infinitely large matrix. The matrix is formed by orthonormal basis expansion, and its element is the overlap integral of the orthonormal basis functions and the Fredholm kernel. If circular aperture is implemented, then it is rigorously proven in this work that the eigenmodes possess certain topological charges (i.e., they are the OAM modes). These OAM modes have specific radial beam profiles, which have been optimized to minimize the power loss and the inter-mode crosstalk. While the traditional OAM beams, such as the Laguerre-Gauss (LG) beams, suffer significant energy loss and inter-radial-mode crosstalk, the optimized beam profiles will remarkably reduce the penalties.
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Free space mode division multiplexing, e.g. orbital angular momentum (OAM) multiplexing, is regarded as the emerging technology for the future optical communication systems [1,2]. Theoretically speaking, it is expected to modulate infinite number of spatial modes to increase the transmission capacity. However, the capacity improvement is limited by two factors: the aperture size constraint and the atmosphere turbulence [2–4].
For the mostly used circular aperture, the angular modes [5–12] and the radial modes [13–16] are considered in multiplexing. The angular modes with different topological charges, i.e. the OAM modes, maintain the orthogonal property after transmission even if the transmitter/receiver have finite aperture size. However, the beam sizes of those modes (e.g. the lowest radial order LG modes with different topological charges) increase with the increase of the topological charges and the largest topological charge is limited by the transmitter/receiver aperture size constraint . Also, the limited aperture size causes power loss for the transmitted OAM modes and degrades the system capacity [2–4]. One of the ways to circumvent the difficulty is to multiplex the radial order modes with specific topological charges [13–16]. It was shown in  that the optimal spatial modes for the finite circular aperture will be the OAM modes with their radial distribution functions as the generalized prolate spheroidal functions. By the beam engineering components like spatial light modulators (SLMs) [12–15], it is possible to realize such optimal spatial modes transmission. However, the modes are only optimal for the ideal free space propagation with aperture size constraint.
Atmosphere turbulence, which is caused by random index fluctuations, forms the other burden for free space transmission. Phenomena like frog, rain, etc. cause atmosphere turbulence and induce random power loss and cross-talk among the transmission modes [3,4,17–22]. Several ways have been proposed to tackle the problem. One may use adaptive optics to compensate the turbulence induced phase and amplitude distortions [3,4,20–22] or may use a mode set which is more resilient to atmosphere turbulence . Apart from using the conventional beams, beam profile optimization under atmosphere turbulence has been an important topic [23–25]. A theoretical framework was proposed in  to find the optimal modes which can maximize the receiving power under atmosphere turbulence; however, no detailed discussions on the practical examples have been given for the optimal modes evaluation. It not clear whether the optimal modes could be the OAM modes for the popular circular aperture, like the case without atmosphere turbulence . In addition to that, it is non-trivial to calculate the two two-dimensional surface integrals during the evaluation of the eigen modes , and no specific method was discussed in . A. Belmonte and J. Kahn  found the LG modes form the eigen modes under atmosphere turbulence, however, ideal LG modes possess infinite field distribution and they cannot be transmitted/received under aperture constraint without truncation. In , a soft-aperture (the Gaussian attenuation function) is considered to avoid abrupt truncation. While it is useful in the analysis, it is different from the practical 'hard-aperture' with abrupt truncation. Henceforth, despite these pioneering efforts [23–25], the optimal mode set like the generalized prolate spheroidal functions with topological charges  have not been found under atmosphere turbulence with aperture constraint.
In this work, we discuss the optimal spatial modes design under atmosphere turbulence with aperture size constraint. When the most commonly used circular aperture is implemented, it is rigorously shown that the optimal modes possess specific topological charges, i.e. they are the OAM modes with different radial distribution functions. The optimal radial distribution functions can be obtained by orthonormal basis decomposition. It is found that the optimally designed modes suffer less loss and inter-mode crosstalk in comparison with the conventional LG modes. With the reduced loss and crosstalk, it is expected to significantly improve the system performance for the free space mode division multiplexing systems.
In this analysis, it is assumed that the statistical information of the channel turbulence is available at the transmitter side, which is natural and technically achievable.
2.1 Power maximization and Fredholm equation
One may directly start with the Fredholm equation proposed in [3,23] to elaborate the beam profile optimization procedures. In order to offer a rigorous demonstration that the obtained eigen modes will achieve the maximal power transmission efficiencies, we start with the average receiving power maximization problem as demonstrated in .
The input field is denoted as u0. According to the Huygens-Fresnel theory , the output field uL can be calculated by:3]:3]. Furthermore, the square law approximation can be applied to simplify the expression of D :
The ensemble average receiving power can be expressed by:Eq. (4) can be found by the calculation of the Fredholm integral equation [3,23]
where λk is the eigen value. It is non-trivial to find the solution to Eq. (5) because of the two two-dimensional surface integrals. Since the maximization problem for Eq. (4) is mathematically equivalent to Eq. (5), we use Eq. (4) as the starting point of the discussions in the next section.
2.2 Eigen modes of the circular aperture
In practice, circular aperture is the mostly used aperture shape for free space transmission systems. Without loss of generality, the transmitter/receiver apertures are assumed to be circular and they are with the same radii of R. Equation (4) can be further simplified to:Equation (6) can be further simplified by integrating the 0th order Bessel function:
It is worth noting that the function of J1(z)/z in the integral can be replaced by the function of (J0(z) + J2(z))/2 to facilitate the calculation process, where J2 is the 2nd order Bessel function.
Since v0 is a function distributed on the circular aperture with the radius of R, it can be expanded by:
In Eq. (9), vector a has been normalized, i.e. aHa = 1, and the superscript H denotes Hermitian transpose. It should be noted that F is a matrix with infinite number of elements, and one may conclude F as a positive definite matrix because of Eqs. (7)-(9) and therefore all of its eigen values are positive. The elements of F are in the form of:
Using the Lagrange multiplier method, it is easy to infer that the maximal value of Eq. (9) should be obtained by taking a as the eigen vector of F with respect to its maximal eigen value λ0max(e.g. a = a0),
This is equivalent to the problem of finding the eigen function with respect to the Fredholm integral [3,23]. Here we have assumed that a0 has been normalized as well. Therefore, one has found the optimal mode which preserves the most transmitted power under turbulence with the aperture size constraint. In addition to the optimal mode above, one may find the secondary optimal beam radial profiles with the coefficient vectors as a1, a2, a3.... and the corresponding eigen values are:
It can be further derived that the above mentioned eigen modes possess specific topological charges. It is proved as follows. Since:
One may change the variable θ1-θ2 to Δθ in the integral and turn Eq. (10) into:
Equation (14) indicates the element of matrix F is non-zero if and only if the two basis functions Rmnejmθ and Rm'n'ejm'θ in the overlap integral have the same topological charges, i.e. m = m'. Therefore, F will be in the form of:
Therefore, the eigen mode problem of matrix F can be decomposed into many eigen mode problems of sub-matrices Fm. Each sub-matrix has the eigen modes possessing a specific topological charge m. Hence, the eigen modes of F are the OAM modes with different radial distribution functions.
2.3 Radial Orthonormal basis expansion of Fm
Fm is also a matrix with infinite number of elements. Since the input field radial function is a function of radius r within the interval [0, R], it can be decomposed by the complete orthonormal basis qn in the form of :
where α0nk is the nth root of the kth (k = 0 or 1) order Bessel function. k has been chosen to be 1 when m≠0, because one should have radial function (m≠0) to be 0 when r = 0 due to the characteristics of the OAM modes. Substituting Eq. (16) into Eq. (14), one gets the elements for Fm. Finding the eigen vector am with respect to Fm, one gets the corresponding eigen modes with the specific topological charge m.
Here Bessel functions are used instead of LG functions, because they converge much faster within the size-limited aperture. In practice, we need to truncate Fm to get an approximation matrix of it, which is also Hermitian symmetrical and positive-definite. Usually, 15-20 Bessel functions are already enough to represent an arbitrary radial distribution function.
3. Results and discussions
Numerical simulations have been conducted to verify the performance of the proposed optimal modes. The simulation parameters are as follows. The transmission distance is 1000m . The wavelength of the signal is 850nm . The input and the output aperture radii are assumed to be 5cm. The optimal mode profiles are designed according to eigen vectors, which corresponds to the largest eigen values of the matrix Fm. Totally 30 Bessel functions (qn in Eq. (16)) are used to expand the OAM mode radial functions to obtain the optimized profiles. For comparison, two types of LG modes with the radial order p = 0,1,2,3 are used as the input . Without atmosphere turbulence, the first type of LG modes (type I) have the corresponding field distributions as [2,14,17]:17]. It should be noted that in Eq. (17), j has been replaced with -j in comparison with the expression in , because Eq. (1) has used exp(jkz) as the propagation phase in the + z direction . With atmosphere turbulence, Eq. (17) applies when z = 0.
The type II LG modes have the input field distributions as 
The second type of LG modes (type II) differ from the type I LG modes by the additional phase term at the transmitter side, which will help to refocus the beam during transmission. Theoretically speaking, the LG modes will have identical intensity distributions at z = L as those at z = 0 if the transmitter and the receiver have infinite apertures . Using Eq. (19) and L = 1km, one may calculate the initial beam waist as 1.64cm. The type II LG modes will have identical beam waists at the transmitter but smaller beam waists at the receiver comparing with the type I LG modes.
Both types of LG modes propagate the same distance of 1000m with the same transmitter and receiver radii of 5cm. The aperture size is chosen based on the results in the references [2,3] without loss of generality. It should be noted that the choice of the aperture size does not affect the conclusion in this work.
The turbulence strength Cn2 varies as 5 × 10−15 m-2/3(weak turbulence), 5 × 10−14 m-2/3 (mild turbulence), and 5 × 10−13 m-2/3 (strong turbulence). Initially, it is assumed to be 5 × 10−14 m-2/3 (mild turbulence).
Since it has been demonstrated in the previous section that the optimal modes possess specific topological charges for circular aperture, we assume the input mode has the form of
where w0m is input radial amplitude distribution function with respect to the topological charge of m. It is assumed in the analysis that the input mode has been normalized so that we have the unit input power:
The output beam, which is also radially symmetrical in the ensemble average sense, has its average radial intensity distribution function as
The optimal beams are studied with respect to different topological charges. First of all, the optimal input and output mode radial intensity distributions with the topological charge of 0 are shown in Figs. 1-3. The red dashed marker in the sub-figures indicates the receiver aperture radius of 5cm. The results in Fig. 1 demonstrate that the designed optimal beam profiles exhibit quite different intensity distributions in comparison with those of the LG modes in Figs. 2-3. Although the designed beams have 1, 2, 3, and 4 rings as the LG modes, the beam waists and the peak values for the four designed modes differ significantly. For the output patterns, there are no nulls as the input patterns because the random fluctuations by the turbulence have the intensity averaging effect.
A quantitative calculation of the power transmission efficiency is performed, which is defined as
The calculated results show that the optimal mode profiles in Fig. 1 have the power transmission efficiencies as 89.27%, 85.55%, 78.81%, and 69.23%, which indicates that little power has been lost during the transmission despite atmosphere turbulence and the finite transmitter/receiver aperture size.
As a comparison, the results of the type I and the type II LG modes with the radial order p = 0,1,2,3 are shown in Figs. 2-3. The corresponding power transmission efficiencies are 84.29%, 72.82%, 62.54%, 53.79%for the type I LG modes, and 87.21%, 81.34%, 75.52%, 69.14% for the type II LG modes respectively. Therefore, the optimal beam profiles show superior performance in comparison with the type I LG modes and the type II LG modes in terms of power preservation. In addition to the power loss, the designed optimal modes maintain the orthogonal property in the ensemble average sense, and therefore, they reduce the inter-mode crosstalk. This will be illustrated in the latter paragraph.
Further demonstration of the input and the output beam radial intensity profiles are plotted in Figs. 4-6 with the topological charge m = 5. The corresponding power transmission efficiencies are 78.51%, 67.33%, 54.39%, 40.98% for the four optimally designed modes, 56.62%, 48.55%, 39.94%, 27.77% for the four type I LG modes, and 72.58%, 66.01%, 52.81%, 30.33% for the four type II LG modes respectively. The type I and the type II LG modes exhibit much lower power transmission efficiencies when m = 5 in comparison with the proposed optimal beam profiles. It can be seen from the intensity distributions that the designed modes have some 'irregularity' (e.g. Figure 4(d)) in comparison with the LG modes, which is the optimization result of power preservation.
Modes with the topological charges of 1-4 are also evaluated for their power transmission efficiencies. The results are listed for the optimally designed modes, the type I LG modes, and the type II LG modes in Tables 1-3. For the readers' convenience, the cases for m = 0 and m = 5 are also included. The radial order modes for the optimally designed mode family are labeled as mode 0 to mode 3 because they cannot be distinguished by the radial order p like the LG modes. It can be seen from the tables that the optimally designed modes show quite remarkable improvement in terms of power transmission efficiencies for different topological charges. If m is negative, due to the rotational symmetry, the modes should have the same power transmission efficiencies as the modes with positive topological charges.
In addition to the power transmission efficiency, the optimally designed beams improve significantly in terms of inter-mode crosstalk, which is defined as
where i and j stand for the ith and the jth modes at the receiver. Although the crosstalk is more useful with its averaged absolution value, the ensemble average of Eq. (24) can also be beneficial for the analysis, particularly when the turbulence is weak. The other reason to choose the ensemble average of Eq. (24) in this work is that the correlation model can deal with this quantity, while the averaged absolution value of Eq. (24) must be analyzed by other approaches and it is beyond the scope of this work.
It is demonstrated in the discussion of Eq. (14) that if and only if the two input modes possess the same topological charges, Eq. (24) will be non-zero. Hence, the ensemble average crosstalk only exists between radial modes with the same topological charge. To demonstrate the strength of crosstalk, the absolution value of the ensemble average crosstalk is shown in Tables 4-9.
Tables 4-5 demonstrate that the optimal beams have zero ensemble average inter-mode crosstalk, indicating that they are orthogonal in the ensemble average sense. Tables 6-7 and Tables 8-9 illustrate quite significant average inter-mode crosstalk for the type I LG radial modes and the type II LG radial modes due to the aperture size constraint, especially when m = 5. This indicates that the LG radial modes become non-orthogonal under aperture size constraint in the ensemble average sense, because the truncation of the LG modes outside the transmitter/receiver aperture introduces crosstalk among the modes. Hence, it can be expected that the optimal beams have better performance in terms of channel crosstalk under turbulence. Particularly, when the turbulence becomes weak, the ensemble average crosstalk becomes close to the actual crosstalk for each channel realization and the optimal beams will achieve nearly zero inter-mode crosstalk.
Finally, the robustness of the proposed optimal modes' performance under different turbulence strengths and propagation distance is investigated.
Firstly Cn2 changes to 5 × 10−13 m-2/3 and 5 × 10−13 m-2/3with the results summarized in Tables 10-13. It can be seen from the tables that power preserving property for the optimally designed mode profiles is well kept under different turbulence strengths. The improvement becomes prominent for the fourth mode when m = 5, because this mode is more severely impacted by the aperture size constraint in comparison with other modes. It should be noted that the change of the turbulence strength should result in different optimal beam profiles. Therefore, proper monitoring of the turbulence strength  is required to generate the optimal modes by the programmable beam engineering techniques [12–15].
Secondly, the optimal beam profile propagation robustness is tested with respect to the different distances of 975m, 1000m, 1025m. It is known from Eq. (14-15) that the optimal beam is L dependent. Therefore, measurement of the propagation distance is necessary before the optimal beam transmission. However, the designed optimal beams are resilient to small propagation length deviations. With the same optimal beams designed for the distance of 1000m and the turbulence strength Cn2 = 5 × 10−14 m-2/3, the power transmission efficiencies only change by a small portion if the distance changes by +/− 25m. The results are summarized in Table 14.
In summary, we propose a set of optimal modes for free space mode division multiplexing under atmosphere turbulence with aperture size constraint. The optimized modes suffer the least power loss and maintain orthogonality in the ensemble average sense at the receiver side. Significant performance improvement is observed through the numerical simulations. The results in this work can be beneficial for the future free space mode division multiplexing system design and optimization.
National Natural Science Foundation of China (61775168); the Shanghai Natural Science Foundation (16ZR1438600).
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