Cite this Article
Cui Xiao-zhou, Yin Xiao-li, Chang Huan, Zhang Zhi-chao, Wang Yong-jun, Wu Guo-hua. A new method of calculating the orbital angular momentum spectra of Laguerre–Gaussian beams in channels with atmospheric turbulence. Chinese Physics B, 2017, 26(11): 114207  
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A new method of calculating the orbital angular momentum spectra of Laguerre–Gaussian beams in channels with atmospheric turbulence
Cui Xiao-zhou1, 2, Yin Xiao-li1, 2, 3, †, Chang Huan1, 2, Zhang Zhi-chao1, 2, Wang Yong-jun1, 2, 3, Wu Guo-hua1, 3
School of Electronic Engineering, Beijing University of Posts and Telecommunications, Beijing 100876, China
Beijing Key Laboratory of Space-Ground Interconnection and Convergence, Beijing University of Posts and Telecommunications, Beijing 100876, China
State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876, China

 

† Corresponding author. E-mail: yinxl@bupt.edu.cn

Project supported by the National Natural Science Foundation of China (Grant Nos. 61575027 and 61471051).

Abstract

Studying orbital angular momentum (OAM) spectra is important for analyzing crosstalk in free-space optical (FSO) communication systems. This work offers a new method of simplifying the expressions for the OAM spectra of Laguerre–Gaussian (LG) beams under both weak/medium and strong atmospheric turbulences. We propose fixing the radius to the extreme point of the intensity distribution, review the expression for the OAM spectrum under weak/medium turbulence, derive the OAM spectrum expression for an LG beam under strong turbulence, and simplify both of them to concise forms. Then, we investigate the accuracy of the simplified expressions through simulations. We find that the simplified expressions permit accurate calculation of the OAM spectrum for large transmitted OAM numbers under any type of turbulence. Finally, we use the simplified expressions to analytically address the broadening of the OAM spectrum caused by atmospheric turbulence. This work should contribute to the concise theoretical derivation of analytical expressions for OAM channel matrices for FSO-OAM communications and the analytical study of the laws governing OAM spectra.

PACS: 42.68.Bz;42.68.Ay;42.25.Dd
Keyword:orbital angular momentum (OAM) spectrum;atmospheric turbulence;free-space optical (FSO) communications
1. Introduction

Over the last 20 years, free-space optical (FSO) communication systems have been developed in an attempt to satisfy the enormous demand for data capacity, and thus, interest in orbital angular momentum (OAM) characteristics is growing.[1,2] Beams with different OAM number l, which is an arbitrary integer, are orthogonal to each other, which leads to the possibility of enlarging the channel capacity in OAM-multiplexed communications.

A practical implementation of a simple FSO-OAM system was reported in 2004.[3] Wang experimentally demonstrated the potential of OAM multiplexing for high-speed communications.[4] Several outdoor demonstrations have also been reported,[5,6] and an FSO-OAM system with a communication distance of 143 km has been realized.[7] All of these works confirm the feasibility of FSO using OAM multiplexing. However, implementations of this strategy face many difficulties, such as atmospheric turbulence, beam divergence, and offset between the transmitters and receivers,[8–13] which can induce the broadening of the OAM spectrum and degrade the performance of the FSO–OAM communication systems. Several previous works addressing beams traveling through atmospheric turbulence have proposed a group of models for describing atmospheric characteristics.[14–23] The OAM spectrum measures the proportion of power remaining in a transmitted channel and crosstalk with adjacent channels in an FSO-OAM communication system, and the OAM spectrum can be used to formulate a channel matrix, which is important for calculating the channel capacity. Thus, studying the theoretical broadening of OAM spectra is necessary.[12] One common method used to calculate the OAM spectrum is based on conjugate spiral phase plates and the principle of orthogonality.[12] In other theoretical studies, probability-based expressions for the OAM spectrum have been derived.[1,10,17–19] In most such studies, based on Laguerre–Gaussian (LG) beams and other types, such as Bessel–Gauss beams, statistical models have been used to analyze and derive precise but complex results for atmospheric channels via model decomposition.[1,10,20–23] The results involve the infinite integration of Bessel functions, which ensures their accuracy; however, the calculation required is enormous, and further derivation is difficult.

In this work, we mainly focus on the broadening of the OAM spectrum caused by atmospheric turbulence. First, we propose a method called the fixed-radius method (FRM) to simplify the expression for the OAM spectrum of an LG beam in a channel with weak/medium atmospheric turbulence. We also derive the expression for the OAM spectrum of an LG beam in a channel with strong atmospheric turbulence and use the FRM to simplify the expression. Then, we define the overall error e to compare the calculation results of the FRM and the elementary formula method (EFM).[1] Finally, we use the simplified expressions to analytically explain the broadening of the OAM spectrum.

This work is expected to contribute to studies of FSO-OAM communications, such as concise theoretical derivations of the OAM channel matrix. Additionally, the resulting expressions can be used to analyze the laws governing the OAM spectrum under atmospheric turbulence.

2. FRM for OAM spectrum calculation

The most common types of vortex beams that are used to carry OAM are LG beams because LG beams are easily realized. Because of the in-axis symmetry of the distribution, an LG beam is described using cylindrical coordinates, and the field distribution for a radial mode of p = 0 can be expressed as[13] where R l 0 ( r , z ) = 2 π | l 0| ! 1 ω ( z ) exp [ − ( r 2 ω 2 ( z ) ) ] [ 2 r ω ( z ) ] | l 0 | × exp { i ( | l 0 | + 1 ) tan − 1 ( z z R ) − i [ r 2 ω 2 ( z ) ] z z R } , and l0 is the angular mode number. The 1/e radius of the Gaussian term is given by , where ω0 is the beam waist, is the Rayleigh range, λ is the wavelength, and (|l0| + 1) tan−1(z/zR) is the Gouy phase.

Figure 1(a) shows the intensity profile of an OAM beam with p=0. Figure 1(b) shows the corresponding intensity distribution; the shape of the beam is a doughnut-like ring, and most of the energy is focused within the ring. Figure 1(c) presents the intensity distribution after it has been disturbed by atmospheric turbulence. The beam remains doughnut-like in shape despite the perturbation. Figure 1(a) indicates extreme points in the intensity profile. LG modes with a radial index of zero are single-ringed annular modes. The intensity profile of the ring resembles a narrow window because most of the power is concentrated in a narrow annulus. By solving the equation ∂ |ul (r, ϕ, z)|2/∂r = 0, we can obtain the radius of the ring at which the intensity is a maximum[13] r ′ = | l 0 | 2 ω ( z ) .

Fig. 1. (color online) Characteristics of OAM beams: (a) intensity profile of a pure OAM beam, (b) intensity distribution of a pure OAM beam, and (c) intensity distribution of an OAM beam under atmospheric turbulence.

If we fix the radius to r′, the doughnut-like shape becomes a thin ring, and the dimensionality (r, ϕ, z) decreases to (ϕ, z). The communication distance z can often be regarded as a constant; in this case, the dimensionality becomes only (ϕ), and we can easily derive the corresponding equations. Although the simplified equations ignore some of the energy in the beam, most of the energy is concentrated at the radius r′. We will demonstrate the validity of this approach in the following sections.

3. Analysis of the OAM spectrum under atmospheric turbulence using the FRM

In this section, we first review the expression for the OAM spectrum under weak/medium turbulence and simplify it. Then, we derive the expression for the OAM spectrum of an LG beam under strong turbulence and simplify it as well.

3.1. Expression for the OAM spectrum under weak/medium turbulence derived using the FRM

For regions with weak/medium atmospheric turbulence, we can use the Rytov approximation to obtain the complex amplitude of a beam disturbed by the weak/medium turbulence as follows:[1] where ψ (r, ϕ, z) is the complex-phase turbulence perturbation and l0 is the transmitted OAM number. The OAM spectrum will broaden because of ψ(r, ϕ, z). The OAM probability density, which we call the EFM result, is given by[1] C ( l ) = 2 π A 2 ω 2 ( z ) ∫ 0 R [ 2 r ω ( z ) ] 2 | l 0 | × exp [ − 2 r 2 ω ( z ) 2 − 2 r 2 r 0 2 ] × I l − l 0 ( 2 r 2 r 0 2 ) r d r , where , R is the radius of the receiver aperture, In is the modified Bessel function of the first kind of integer n, and is the coherence radius of the spherical wave. Here, z is the transmission distance, and is the refractive index structure parameter, for which one of the most widely used models is the Hufnagel–Valley model[24] C n 2 ( z ′ cos α ) = 8.148 × 10 − 56 v 2 ( z ′ cos α ) 10 exp ( − z ′ cos α / 1000 ) + 2.7 × 10 − 16 exp ( − z ′ cos α / 1500 ) + C n 2 ( 0 ) exp ( − z ′ cos α / 100 ) , where is the refractive index structure parameter that is characteristic of the ground, α is the zenith angle, and v is the square root of the wind speed. Over short time intervals at a fixed propagation distance and a constant height above the ground, it may be reasonable to assume that is essentially constant.[24] Let us fix the radius to ; after normalization, we obtain the following simplified expression for the OAM spectrum:

3.2. Expression for the OAM spectrum under strong turbulence derived using the FRM

Unlike in the case of weak/medium turbulence, in a region with strong turbulence, the normalized complex amplitude of the beam can be expressed as[18] u l , strong ( r , ϕ , z ) = u l , free ( r , ϕ , z ) exp [ ψ x ( r , ϕ , z ) + ψ y ( r , ϕ , z ) ] , where ψx (r, ϕ, z) and ψy (r, ϕ, z) are the complex-phase perturbations caused by large- and small-scale turbulence eddies, respectively.

The most common method used to obtain the OAM probability density of any vortex beam under strong turbulence can be expressed as[18,19] C l = ( 1 2 π ) 2 ∫ 0 2 π ∫ 0 2 π u l , free ( r , ϕ , z ) u l , free * ( r ′ , ϕ ′ , z ) × exp [ − i l ( ϕ − ϕ ′ ) ] 〈 exp [ ψ x ( r , ϕ , z ) + ψ y ( r , ϕ , z ) + ψ x * ( r ′ , ϕ ′ , z ) + ψ y * ( r ′ , ϕ ′ , z ) ] 〉 at d ϕ d ϕ ′ , with 〈 exp [ ψ x ( r , ϕ , z ) + ψ y ( r , ϕ , z ) + ψ x * ( r ′ , ϕ ′ , z ) + ψ y * ( r ′ , ϕ ′ , z ) ] 〉 at = exp [ − 1 2 D ( r , ϕ , r ′ , ϕ ′ , z ) ] = exp [ − 1 2 ( D x ( r , ϕ , r ′ , ϕ ′ , z ) + D y ( r , ϕ , r ′ , ϕ ′ , z ) ) ] = exp [ − r 2 + r ′ 2 − 2 r r ′ cos ( ϕ − ϕ ′ ) ρ 0 2 ] , where ρ0 is the spatial coherence radius in a channel with strong irradiance fluctuations and is given by[18] ρ 0 = ( 0.491 σ R 2 k z { ( 35.046 η x z k η x l i 2 + 35.046 z ) 1 / 6 + [ 1.766 − ( 35.046 η y z k η y l i 2 + 35.046 z ) 1 / 6 ] } ) − 1 / 2 , with and Here, li is the inner scale of the turbulence, and is the Rytov variance. Based on the integral expression[18] ∫ 0 2 π exp [ − i n φ 1 + η cos ( φ 1 − φ 2 ) ] d φ 1 = 2 π exp ( − i n φ 2 ) I n ( η ) , we can combine Eqs. (3), (8), (9), (10), and (11) to derive the expression for the OAM probability density of an LG beam under strong turbulence as follows: β l = ( 1 2 π ) 2 [ A ω ( z ) ] 2 [ 2 r ω ( z ) ] 2 | l 0 | exp [ − 2 r 2 ω ( z ) 2 ] × exp ( − 2 r 2 ρ 0 2 ) × 2 π I l − l 0 ( 2 r 2 ρ 0 2 ) . We can also derive the expression for the OAM spectrum under strong turbulence as follows: P ( l | l 0 ) = 〈 | β l | 2 〉 ∑ m = − ∞ ∞ 〈 | β m | 2 〉 = ∫ 0 R [ 2 r ω ( z ) ] 2 | l 0 | exp [ − 2 r 2 ω ( z ) 2 ] exp ( − 2 r 2 ρ 0 2 ) I l − l 0 ( 2 r 2 ρ 0 2 ) r d r ∑ m = − ∞ ∞ ∫ 0 R [ 2 r ω ( z ) ] 2 | l 0 | exp [ − 2 r 2 ω ( z ) 2 ] exp ( − 2 r 2 ρ 0 2 ) I m − l 0 ( 2 r 2 ρ 0 2 ) r d r . This is the EFM result for strong turbulence. Using the FRM, we fix the radius to r′ and obtain the following simplified expression for the OAM spectrum under strong turbulence:

4. Analysis of the validity of the FRM in channels with atmospheric turbulence

In this section, we demonstrate the validity of the FRM. We consider several OAM spectra, as depicted in Fig. 2, which show the proportion of the power that remains in the transmitted OAM state or leaks into other states via crosstalk. We set the parameters as follows: ω0 = 0.01 m, λ = 1550 nm, z = 500 m, v = 21 m/s and the zenith angle is set to α = π/2, which indicates that the beam is transmitted horizontally, and consequently, equation (6) becomes a function of only z. Considering typical manufacturing limitations and the size of a typical receiver, we set the receiver aperture radius to R = 10 cm.[25] Figures 2(a) and 2(b) present the OAM spectra for different transmitted OAM numbers under weak/medium and strong turbulence, respectively. Because most of the power is concentrated around state l0, only finite terms are considered. We use 21 and 61 states around l0 to calculate the results of Eqs. (7) and (15), respectively. For comparison, we also present the results obtained using the EFM. As shown in Fig. 2, the results of the two methods are consistent, particularly for larger OAM numbers.

Fig. 2. (color online) OAM spectra for l0 = 5, 10, 15, 20 under different levels of turbulence: (a) weak/medium turbulence, ; (b) strong turbulence, .

To quantitatively compare the EFM and FRM results, a dimensionless parameter called the overall error is defined as follows: e = ∑ l = − ∞ ∞ | P EFM ( l | l 0 ) − P FRM ( l | l 0 ) | . We do not need to normalize the parameter e because the sum of the entire OAM spectrum is equal to 1. The value of e lies between 0 and 1, and a smaller e corresponds to more accurate FRM results. Figure 3 shows the overall error versus turbulence for l0 = 5, 10, 15, and 20. We use 21 and 61 states around l0 to calculate the results of Eq. (16) for weak/medium and strong turbulence, respectively.

Fig. 3. (color online) Overall error e for l0 = 5, 10, 15, and 20: (a) weak/medium turbulence, (b) strong turbulence.

None of the e values in Fig. 3 exceeds 0.05. For example, the overall error corresponding to Fig. 2(b-1) is 0.0425, and the results also appear visually identical; furthermore, this figure shows that even the worst situation can be accepted. Thus, the results of the two methods are nearly identical for both weak/medium and strong turbulence. In Fig. 3(a), under weak/medium turbulence, the overall error increases gradually with increasing , and at larger OAM numbers, the overall error is smaller. By contrast, the errors in Fig. 3(b) are nearly ten times of those in Fig. 3(a), and the overall errors remain nearly steady. Thus, the overall error initially increases as increases and remains steady once reaches the strong turbulence range. Therefore, we can conclude from Fig. 3 that the accuracy of the FRM is acceptable under any level of turbulence, particularly in situations with weaker turbulence, when the OAM number is sufficiently large.

Figure 4 shows the overall error versus the OAM number l0 for weak/medium and strong turbulence. Figure 4(a) shows that for a given turbulence, e gradually decreases as l0 increases, particularly under strong turbulence. In the range of small OAM numbers, the overall error remains nearly constant, whereas e drastically decreases as l0 increases in Fig. 4(b). For small OAM numbers (l0 ≤ 5) and strong turbulence, as shown in Fig. 4(b), the overall errors exceed 0.1, and the accuracy is low. This is because when the OAM number is small, the ring over which the beam intensity is distributed is thick, and strong turbulence causes the intensity distribution to deviate from an annulus. Thus, we can conclude from Fig. 4 that when l0 is larger, the overall error is smaller, and the FRM method is not suitable (when e exceeds 0.1) for small OAM numbers (l0 ≤ 5) and strong turbulence.

Fig. 4. (color online) Overall error e of the OAM spectrum calculations in different situations: (a) weak/medium turbulence, (b) strong turbulence.

In conclusion, the proposed simplified expressions for the OAM spectrum are highly accurate, particularly for weak turbulence and large OAM numbers.

5. Discussion

Based on the simplified expressions for the OAM spectrum, we can analytically investigate the rules governing the variation of the OAM spectrum with the OAM number l0 and the turbulence strength.

Interestingly, the results of Eqs. (7) and (15) are similar, except for the expression of the coherence radius. Because I−α (x) = Iα (x),[1] we can write a single simplified expression for the OAM spectrum under either weak/medium or strong turbulence as follows: where m is the OAM number adjacent to the transmitted l0 and x = { 2 r ′ 2 r 0 2 = ω ( z ) 2 | l 0 | r 0 2 , for weak/medium turbulence , 2 r ′ 2 r 0 2 = ω ( z ) 2 | l 0 | ρ 0 2 , for strong turbulence . Equation (18) shows that x is directly proportional to |l0| and inversely proportional to or . The remaining relative power in the transmitted OAM state |l0| can be expressed as Equations (17) and (19) show that when x is fixed, the OAM spectrum will remain the same. Thus, an increase in l0 at a fixed level of turbulence can induce an effect identical to that caused by an increase in at a fixed l0. Consequently, we can obtain the same OAM spectrum by either increasing l0 or decreasing .

According to Eq. (18), if the turbulence and transmission distance are known, which implies that ρ0 or r0 is fixed because x is directly proportional to |l0|, an increase in the transmitted OAM number l0 causes x to increase. From Eqs. (17) and (19), we can see that the OAM spectrum depends on the ratio Im (x)/I0 (x). To show the relationship between l0 and the OAM spectrum at a fixed level of turbulence, we numerically calculate Im (x)/I0 (x) for m = 0, 1, 2, 3, 4, and 5, with and z = 1000 m. The calculation results are shown in Fig. 5. Figure 5 shows that Im (x)/I0 (x) monotonically increases with l0 and when m increases, Im (x)/I0 (x) decreases.

Fig. 5. (color online) The Im(x)/I0(x) for m = 0, 1, 2, 3, 4, and 5.

For Eq. (17) with m ≠ 0, some power leaks into other OAM states. When l0 and the turbulence are fixed, we can compare the power leaking to adjacent OAM numbers simply by comparing Im (x)/I0 (x) because the denominator of Eq. (17) is constant. In Fig. 5, when m increases, Im (x)/I0 (x) decreases. Thus, the relative power in the OAM spectrum is related to the distance of the OAM number from l0; as the OAM number moves away from l0, the power that leaks to it decreases. Thus, in Fig. 2, the turbulence-disturbed OAM spectrum assumes a bell-like shape, being tall in the center and declining on the sides.

According to Eq. (19), the remaining relative power in the transmitted OAM state P(l0|l0) decreases as l0 increases, as shown in Figs. 2(a) and 2(b). We can also consider crosstalk with other OAM states based on Eq. (19), where Im (x)/I0 (x) is included in . Therefore, an increase in l0 will cause P(l0 + m|l0) to decrease. Thus, the curves in Fig. 2 become flatter as l0 increases.

To show the relationship between and the OAM spectrum, we also plot the curves of Im (x)/I0 (x) for different with a transmitted OAM number of l0 = 15 and a transmission distance of z = 1000 m in Fig. 6.

Fig. 6. (color online) The Im (x)/I0 (x) for m = 0, 1, 2, 3, 4, and 5.

In Fig. 6, Im (x)/I0 (x) monotonically increases with increasing and decreases as m increases, which is identical to the behavior observed with increasing l0 and m in Fig. 5. As seen from Eqs. (17) and (19), for a fixed l0 and a fixed transmission distance z, r0 and ρ0 vary inversely with . Thus, an increase in causes Im (x)/I0 (x) to decrease, implying that an increase in turbulence will cause the entire OAM spectrum to become lower in peak power and exhibit increased broadening.

6. Conclusion and perspectives

We propose the FRM for deriving simplified expressions for the OAM spectrum. We fix the radius of the OAM beam to the extreme point of the intensity distribution to make the expressions more concise and reduce the amount of calculation. Then, we define an overall error e to measure the accuracy of the FRM results. Based on this error calculation, we find that the FRM has high accuracy, particularly for large transmitted OAM numbers. However, the overall error is relatively large when the transmitted OAM number is small (l0 ≤ 5) and the turbulence is strong. Finally, we demonstrate that the broadening of the OAM spectrum caused by atmospheric turbulence can be studied analytically using the proposed simplified expressions. Moreover, a concise expression without infinite integration will be important for future derivations of channel matrices and simulations of multiplexed FSO-OAM systems.

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